"Getting a bank account is fairly straightforward, and most don't require a SSN."
I don't know about this. It's pretty much impossible to open a bank account without a SSN or TIN.
You can open the account without an SSN. However most banks will generally require you to provide a valid SSN or TN with a certain period of time otherwise they will close the account.
Generally if you are in the US legally you will have one of those numbers.
I got my bank account before getting my SSN. They ask for passport and maybe some other documents (e.g. I-20 for F-1 Visas) but certainly getting a bank account without SSN is possible.
