I replicated your test transfer several hours before you mentioned Yodlee. Only later, an hour or two after you subsequently named it as the third party did I try again. I tried again because as you noted Vanguard started providing alternative paths.
In any case, had you been able to find/follow the "Transfer an account" link, it would have taken you to:
https://personal1.vanguard.com/mmx-move-money/welcome
The problem may be with the browser used. With the first couple of browsers I tried, the Buy and Sell page is as I described, including a "Transfer an account". With a Chrome browser, I get what's probably an older version of that page. As you said, without a "Transfer an account" link.
Using the move money link above, the Chrome browser still leads one to the Yodlee prompt, while other browsers get to the pages I previously described. Best suggestion is to try different browsers until one of them works for you.
(BTW, even from the old Buy and Sell page, the Open an Account link will take you to the same place, i.e. the move money page. For all the good that does.)

On wonders if Vanguard is listening.
As BaluBalu had not (until an hour ago) identified the third party, and because I did not recall being prompted by Vanguard to provide Yodlee any information, I started a test transfer. Sure enough, in the process a Vanguard page identified Yodlee and asked for login info for the "from" account.
I just tried reproducing that and Vanguard's site took me through a different sequence.
From MyAccounts (drop down) By & Sell, I followed the "Transfer an account" link. Next page, "Start your transfer or rollover". The next page has a radio button for type of account. I selected Roth IRA.
The next page asks whether the assets are in one's possession or at another firm (radio button). Moving on, the next page reads:

Things you'll need

• The name of the other institution.
  
• Your account number.
  
• The type of account you're transferring (e.g., IRA, individual, joint).
  
• The types of investments in the account (e.g., stocks, bonds, mutual funds).

Transfer process

1. Complete our transfer form (10 minutes)
   Give us some information about the outside account you want to transfer, and we'll guide you through the process.
   
2. We'll initiate the transfer
   Your investments will transfer in kind from your outside account to your new Vanguard account. Depending on your institution, we'll receive your assets electronically or by mail. [Emphasis added.]
   
3. Watch for a confirmation
   We'll email you status updates throughout the process as well as a confirmation when the transfer is complete.

Nothing about logins. Indeed nothing else is required.
After providing the requested info, I get to a page that recaps the information submitted (sending brokerage, account number, assets to be transferred). The step is labeled: Review and e-sign.

We did an IRA transfer from Fidelity to Vanguard a year ago that went via ACAT, just as Observant1 described. When I spoke with Vanguard I was told that the medallion guarantee requirement originates with the sending brokerage. If they don't require the guarantee everything can be done electronically. Vanguard specifically commented that they have a good working relationship with Fidelity.

When I transferred assets to Vanguard, the Medallion Signature Guarantee requirement originated at Fidelity.
Although this caused a delay, the process helps to prevent fraud/theft which I can appreciate.
Here's a good primer on Medallion Signature Guarantees.

@BaluBalu - I’m reluctant to offer advice on a sensitive matter like this. But here goes …
First, in transferring some funds from my credit union to Fidelity for the first time (about a month back) a message on the screen asked for my credit union user name and password. It appeared at the time to me that this was an added layer of security required by the credit union before they would process the request from Fidelity. I supplied the requested information and the transaction was completed w/o any hassle. FWIW - I’d linked my bank several weeks earlier w/o having to do this.
Second - While I don’t recall any reference to a “third party”, I’m wondering in your case if that might be an added safeguard - so that VG would not have 100% control over the transaction. In other words, I suspect it is a trusted 3rd party serving as added / independent security measure. I could be wrong.
(Delete)
*Another thought - Had I been worried about having provided the user name & password I’d have immediately logged in to my credit union account and changed them. Never got around to it. Not a bad idea.

My TDA accounts are grandfathered for a commission rate of just $15 to purchase non-NTF mutual funds such as Dodge and Cox or Vanguard. I'm waiting to see if Schwab honors that rate. (I suspect that I would lose that benefit if I transferred my TDA accounts to Schwab before the integration.)

May be call Schwab and ask the question. If they agree to honor, then transfer now to lock in; otherwise, enjoy the benefit for another 2 years. They may honor now not to lose you to a competitor who may offer large incentives to get your accounts.

Better than Bitcoin ….. From this week’s Barron’s (11/1)
“Bitcoin investors are probably thrilled with the coin's 50% gains over the past few weeks. But that's nothing compared with the Squid Game token that popped up this past week. Pegged to an online game inspired by the hit Netflix series, the “play to earn” coin rocketed nearly 5,000% over three days, going from 12 cents to $6. It's now worth $475 million, according to CoinMarketCap.”

We did an IRA transfer from Fidelity to Vanguard a year ago that went via ACAT, just as Observant1 described. When I spoke with Vanguard I was told that the medallion guarantee requirement originates with the sending brokerage. If they don't require the guarantee everything can be done electronically. Vanguard specifically commented that they have a good working relationship with Fidelity.
Regarding the hoops Level5 jumped through, the rules are different for employer-sponsored plans. With those plans, the withdrawal must be initiated on the 401k/403(b)/457 side. The assets usually have to be converted to cash. The only exception I have seen, and this is with multiple financial institutions, is that if you're transferring assets between an employer plan and an IRA run by the same institution, e.g. TRP, Fidelity, etc., then the transfer can be in-kind.
Many if not most of these plans today still send the cash only via paper check. The only way I've found around this problem is to use the exception above: transfer assets electronically from plan to a temporary IRA managed by the same institution if possible. Then have your target IRA do a customary electronic pull from the temporary IRA.

Most of Vanguard's index funds offer ETF class shares, so one can avoid using the Vanguard platform to invest in those funds. However, none of Vanguard's allocation index funds (target date, target risk, or "traditional balanced") or tax managed index funds have ETF class shares.
Oddly, neither does VFTAX even though Vanguard has three ETF-only ESG funds.
https://investor.vanguard.com/investing/esg/
I agree with you about the last three firms you mentioned. I find Vanguard serviceable, the others not so much.
E*Trade - ran my employer's stock purchase plan. Refused to let us sell shares of a new company created in a spinoff. Forced us to open a new retail E*Trade account or to pay E*Trade to get the shares out. I didn't appreciate being held captive.
Merrill Edge - here's the thread I started a couple of years ago; 'nuff said.
TIAA - impossible to navigate, reducing offerings. They used to offer a simple, vanilla brokerage IRA. No longer. Now all they offer is an IRA annuity with a brokerage window. Since it's an annuity you can't just have another brokerage pull your assets (not even cash). And to open this oddball IRA you have to be "eligible".
https://www.tiaa.org/public/pdf/eligibility_flyer_external.pdf

1 hour every Jan, April, July, October.

Vanguard Customer Service sucks. Historically. I suffer them because I consider them safe and use their funds for indexing.
Oh and another thing. They sense of "rounding" is illogical regarding how many shares they give you for whatever price even when you are investing nice whole $ amounts e.g. $1000.
You want customer service, my top pick amongst the "big guys" is Fidelity. Schwab comes next (while it hasn't yet translated over to TD Ameritrade).
PS - If you think Vanguard is bad, you haven't experienced E*Trade (now Morgan Stanley) or Merrill Lynch (barf), TIAA-CREF (someone just shoot me)

Here are notes related to my 2019 account transfer to Vanguard.
Changes may have been implemented since then...
Will I pay any fees for my transfer?
Vanguard doesn't charge any transfer fees. However, you'll need to contact your current financial firm to see if it charges transfer fees for closing an account.
If you want to move money from a CD (certificate of deposit) that's held in a bank account and the CD hasn't yet reached its maturity date, check with your firm to see if you'll be charged any withdrawal fees or penalties. Bank CDs will be liquidated at the time of the transfer to a Vanguard account.
Is there any paperwork I need to fill out?
Many large financial firms are enrolled in the Automated Customer Account Transfer Service known as ACAT (or a similar service). The majority of ACAT transfers can be completed entirely online, with assets sent to Vanguard electronically. Some cases, such as partial transfers and changes of ownership (e.g., a joint account to an individual account) may require paperwork. If forms are required, we can prefill some of the information for you.
Some asset transfers require a Medallion signature guarantee from a bank or another financial institution. A bank officer, trust company, or member firm of the U.S. stock exchange can grant this service. (Notary publics can't provide it.) If the firm you're transferring from requires a signature guarantee, check with your bank or a financial institution near you to see if they'll provide this service.
How long will the transfer take?
The fastest transfers take approximately 7 business days. They occur when firms are enrolled in the Automated Customer Account Transfer Service known as ACAT (or a similar service), when assets are moved "in kind" (or "as is"), and when the owner's name is identical on the "to" and "from" accounts. When assets are moved "in kind," it means there's no selling or buying involved, and no gain or loss is recognized as a tax consequence.
A firm that doesn't provide automated transfers may have you fill out paperwork and will send us a check once it receives your documents in good order. This may take 4 to 6 weeks.
Assets held in a money market fund are cashed out before they're moved to the money market settlement fund in your Vanguard account, so they may arrive after your other assets.
Many brokerage CDs (certificates of deposit), limited partnerships, hedge funds, and low-priced securities can't be transferred in kind. If you hold any of these investment in your account, we may contact you to discuss your options.

Full Episode Oct 30, 2021:

My TDA accounts are grandfathered for a commission rate of just $15 to purchase non-NTF mutual funds such as Dodge and Cox or Vanguard. I'm waiting to see if Schwab honors that rate. (I suspect that I would lose that benefit if I transferred my TDA accounts to Schwab before the integration.)

@Mark. Thanks much for the excerpts! And I agree with your explanation.
1st paragraph is accurate in that during normal times an overheated equity market (and economy) are likely to invoke Fed tightening. Eventually (late in the tightening cycle) the market cools off. Than, as the economy cools, the Fed lowers rates in hopes of reigniting growth. That’s a normal cycle. The question remains as to whether the present (as well as the past decade or two) represent “normal”.
2nd paragraph seems an accurate summary of the Fed’s position as they have relayed it through various means (press conferences, meetings’ minutes, statements). No quarrel here. That’s what they’ve said.
Than comes the following inference: “Given this data and the fact that the overall market has shown to perform best when rates are stable, it appears likely that stocks can do considerably well until then." Here, we’re dealing in the realm of probability.
I have no crystal ball. If we’re in normal times, and if the Fed acts going forward in the way it has indicated it will, and if probabilities based on past patterns hold true …. off to the races!

From the article:
"The tables show the stock market as a whole did best when rates were steady, as contrasted with what you might expect, with an average annualized return of 26.73% during four such periods. Surprisingly, it did the worst when rates were falling with an average annualized return of -3.72, and with an average annualized return of 10.89 when rates were rising!"
Equal's - by the time the Fed reacts, raising or lowering, the results have already been generally cooked in.
Implications for Investors
"Right now, even though the Fed is highly likely to quite soon begin phasing out its bond purchases, called quantitative easing, that is not the same as actually raising rates. Since the Fed Chairman has repeatedly stated the Fed is not expected to raise rates until those purchases have ended sometime later next year, we can assume that rates will remain stable until then, as they have been since the last series of cuts ended on March 15, 2020. Given this data and the fact that the overall market has shown to perform best when rates are stable, it appears likely that stocks can do considerably well until then."

According to the Oct 21, 2021 issue of "Investment News," Schwab hopes to move the individual T.D. Ameritrade accounts into Schwab by "the second half of 2023." That's still around two years away!
I had expected it would be faster.

RPHYX of course now closed.
Here's the August 24, 2021 supplement to the fund prospectus:

August 27, 2021 (the "Revised Closing Date") ...
After the Revised Closing Date, the following eligible investors may also open new accounts:
· New shareholders may open Fund accounts and purchase shares directly from the Fund (i.e. not through a financial intermediary).
· Any trustee of RiverPark Funds Trust, or employee of RiverPark Advisors, LLC or Cohanzick Management, LLC, or an investor who is an immediate family member of any of these individuals.

https://www.sec.gov/Archives/edgar/data/1494928/000139834421016711/fp0068207_497.htm

reserving the right to reject orders exceeding ...
You were given imprecise information. Fidelity, like most fund sponsors, puts in boilerplate allowing them to reject any purchase, including a purchase via an exchange if they feel it would disrupt the fund. But not sell orders. If they did, the funds would no longer be classified as OEFs.

An open-end fund is required by law to redeem its securities on demand

https://www.sec.gov/rules/proposed/2015/33-9922.pdf
Based on the purchase dollar limit you were given for FCNTX, and the limit that I actually hit on a very new and very small Fidelity fund, it looks like Fidelity sets its fund limits at 0.1% of AUM. (M* shows FCNTX as having $139.5B, or roughly 1,000x the purchase limit.)
Regarding redemption in-kind, Fidelity (or any fund company) would distribute securities owned by the fund. Obviously if the fund were to sell some securities just to purchase other ones to hand you, it might as well hand you the cash since that would be no more disruptive.
As it constitutes 10.65% of the fund's portfolio, I'd expect you to get a ton of FB.
According to the latest semiannual statement, Fidelity Contra redeemed 293,065 FCNKX shares in kind, worth $5,071.454. It does happen.

I transferred my Roth IRA from Fidelity to Vanguard in 2019.
The first three comments listed in the link below provide details.
Link