Yes, you are correct, you only really need proof of address once you start offering credit (running hard credit check).

And you're also correct in terms of AML - once a person crosses over a certain threshold or raises and risk alarms, they'll be required run further AML checks. Eg. if you were to signup to Monzo and then put 50 grand in there from a foreign account.

What Monzo has to do since they are new, is keep a squeaky clean record with the authorities. This is why Monzo usually blocks or even bans anyone that buys Cryptocurrency eg: https://www.reddit.com/r/monzo/comments/avdw8o/notice_regard...

Source: I work in Crypto