I was actually talking about my own experience canceling a gym membership. As mentioned, I used a Privacy card, which is locked to one vendor. You can turn off that specific card, and nothing else is affected. That’s how I canceled my own gym membership, when I couldn’t do it in person, because the gym was closed during Covid.
I mentioned my email service because it has a very similar feature to Privacy, I’ve also used it when going as a guest to a different gym, and OP mentioned getting constant emails from the gym. This relates to both their and my experience.
Blocking emails doesn’t stop the payments either.
That’s why I was talking about Privacy. Turning off a Privacy card does stop the payments.
as if that’s the simple and obvious solution, with no other detrimental effects
If you’re unfamiliar with Privacy, it’s basically like a disposable email service, but for credit cards. You only use a Privacy card with one vendor, so there aren’t any downsides to turning it off.
If you think that my comment is unrelated to OP’s situation, then I really don’t know what to tell you. Maybe try canceling a gym membership and see for yourself.