2) If a vendor is "accidentally" charging you and you're actually in the right, chargebacks fix that instantly.
3) If you have an actual (multi-)year contract with a gym, cancelling your card doesn't get you out of that contract. Charges can often still be made to a closed card, you'll get bills from either the gym or your credit card provider, and it will go to a collections agency if you don't pay (and again, knock down your credit score).
I really hope you don't actually do this because you'd be harming yourself for zero actual benefit.
You can only request a chargeback, and chargebacks only work sometimes. I've done maybe 10 in my life, and I think 1 out of those 10 didn't go through -- the bank sided with the merchant. And I think if you do too many, the banks get pissed off. So depending on chargebacks is not a panacea.
And I'm not talking about reneging on my responsibilities or contracts that I've entered into. I always resolve those.
Furthermore, vendors with high dispute rates get shut down pretty quickly by payment processors. Your behaviour is harming you far more than it's helping.
This is a feature: You update your details and don't have to individually talk to every single service out there that use the old details.
Of course ideally banks would have a way to let the users control these types of features but, you know, banks suck.
Your behavior isn't a bug but it is atypical. Your workaround to an infrequent problem just requires a new step: closing and opening new credit card accounts.