I’ve been paying Claude in cash by showing it a picture of $5 bills as I burn them. It says my account is good.

Doesn't say anything about citizenship though. There are plenty of US residents who are not citizens. And a lot of people abroad appear to use US billing address credit cards -- in my last company we had hundreds of people with the same US billing address who appeared to be managing Africa-focused businesses and used IPs that matched that.

No requirement that you pay with your own CC, however.