Many brokerages provide cash management services as if they were real banks. For example, Fidelity accounts offer checking ant debit card services. But those services are provided by UMB Bank (routing number 10120
5681is associated with
UMB Bank, KS).
Fidelity says that its "accounts are
considered a checking account for direct debit purposes."
A few institutions will link only to real banks, not to these pseudo "bank" accounts. I don't have any problems linking Treasury Direct to such accounts. So it doesn't look like Treasury Direct is one of those few picky institutions.
On the other end of the spectrum are real brick and mortar banks with lots of branches. Such as Chase, or TD Bank that is "one of the 10 largest banks in the U.S" and "a subsidiary of The Toronto-Dominion Bank of Toronto, Canada". They are FDIC insured, unlike brokerage accounts masquerading as checking accounts.
https://www.tdbank.com/aboutus/investor_relations.htmlWhen it comes to Schwab, Schwab Bank is a real, FDIC-insured bank, separate from but affiliated with the broker-dealer Charles Schwab & Co. "Affiliated" here means that they are both owned by the same parent company, The Charles Schwab Corporation.
https://www.schwab.com/savingsBut the Schwab One brokerage account (as opposed to the
HY checking account) appears to operate as a pseudo "bank" account. The account disclosure says that "Checking account and Debit Card services provided by the Bank". And it defines "Bank" to be "BNY Mellon Investment Servicing Trust (IST) Company and/or its affiliates, the entity responsible for administering the Bank Services."
https://www.schwab.com/legal/schwab-one-account-agreementThen there are internet-only banks (e.g. Ally Bank). In theory that shouldn't make any difference in the context of electronic transfers as opposed to walk-in services. One doesn't walk into a bank to set up a domestic EFT link.