Is there a way to use an app similar to Privacy to set up a "burner bank account"? Or do you physically go into a bank, credit union, etc. and sign up for a new account as a new member?
And most banks that I'm aware of will let existing customers create new accounts with distinct routing/account numbers online.
Not in the US. Especially unique routing numbers. Maybe you mean virtual credit card numbers.
A bank will only have one or a small number of routing numbers.
My point was that most banks will let existing customers open new accounts online. This will create a new routing/account number combination that you can share with the service you don't trust enough to share your primary account with.
You can also add a (new) debit card to the new account, and some providers let you create virtual cards at will. But that's not what I was talking about.