Even if I'd simply "forgotten" to cancel, the prohibition on charging me for the months I didn't use it would've made this a non-issue. Hell, I'd probably still be a customer today, now that I've long ago moved back to a city with Anytime Fitness locations.
But you didn't. You clearly stated that the burden of cancelling was too high: " The gym staff admitted to remembering that I attempted to cancel because I was moving to a place with no Anytime Fitness locations; they refused to let me cancel my contract early without me showing them my new lease, which I didn't have yet"
This is the root of the problem. Not the "prohibition to charge for services you've subscribed to but don't use".
No, the problem would be moot if the cancellation process was as easy as the sign up process. And I think the US finally got that law