most credit cards allow you to create a temporary card number. Create one, set it to be the billing method, and then revoke it. crazy that we need to resort to that sort of thing, but it does work!
Canceling your credit card doesn't magically get you out of owing money that you're contractually obligated to pay. It might get them to eventually cancel your service for non-payment, but it's not a guarantee. They might just keep billing you until it's worth thousands and then mess up your credit or pursue you in court for payment.
I've talked to my rep about the idea of "credit card cancellation": the idea that you should be able to go through your bank (or you credit card provider's web app) go to a recurring charge and click "cancel" from there. I'm pretty sure most major credit card companies would be on board; what's stopping them is the legal thicket of contracts that are in the way. What the CC providers need is a clear framework from the legislature to support them. The FTC ain't enough.
Actually that is probably one thing that credit cards could do without legislational changes, by modifying their merchant agreements. Businesses would be salty about it, but the value of being able to accept credit cards would mean they'd have to suck it up and accept it. They'd have to put something about how by accepting cards, they agree that that agreement supersedes any other and that a payment cancelation or chargeback constitutes acceptance of cancelation for whatever service was being paid for by that card or something along those lines.
It might be worth trying if you keep a close eye on it, because there's a chance they cancel you after a couple of non-payments.
Does this fix whatever method companies use to continue billing you monthly when you are issued a new card because the old one was lost/expired/etc?
I have several credit cards, none of the providers allow me to create a temporary number. Plus, one wouldn’t be enough because you’d need one for every vendor you might want o cancel in the future.