Because the very purpose of a stablecoin is that one person buys the stablecoins with real USD; okay, you can KYC with them. But then once they have those stablecoins they go off and use them - to pay someone else for something - like some Bitcoin or something. Possibly something illegal.

This is not someone you have a direct relationship with. And now those stablecoins ‘belong’ to that new person.

Then that person uses the stablecoins to pay another person for something else, and eventually a completely different person can now come along and go through your KYC process and cash out the coins.

All above board and legitimate.

Except in the middle there’s a part where you might actually be acting as a bank for an international drug cartel or a sanctioned Russian oligarch. You just can’t be sure.

Agreed, but as long as the people you interact with are above board, it's on them if they use it wrong, no? You could make the same argument for physical treasury bonds, or gift card codes, or generally any interaction with crypto currency, but I don't see any bank having an issue with that.