So, I got very excited about this, but it seems that banks are expecting "bank integrator" aka companies, and not giving access to end users :( If any knows of a bank that has API access in the US do share!

You might find luck using companies targeting algo trading. A lot of companies allow use of the account more like a checking account (eg interactive brokers). They have an API and also allow different logins to have different authorizations.

Mercury Bank has nice, friendly APIs. I believe they only do business bank accounts, though.

But I think I remember seeing that some of Capital One's bank account offerings have some customer facing apis.