They will just continue attempting to collect money as per the contract you signed, and then send you bill to collections when they can't.
Edit: Credit card companies typically require/ask you to dispute with the merchant and attempt to do get a refund first before they will chargeback. If you try, and the gym can point to contract, you'll lose the dispute either way. Getting your credit card number changed stops the gym from charging you, but you'll still owe them money and you'll typically find out when you start getting calls from a collections agency.
It's not "blaming the consumers" for expecting people to follow the terms of contracts they sign. I never had a Gold's Gym membership for exactly this reason - their cancellation terms were onerous, I wasn't interested in complying, so I never signed and never gave them any money.
If you say "well, I don't want to do that, but I'm just going to sign this anyway then do a chargeback because that's easier" them yes, you deserve to be blamed, you deserve to be shamed, and you should have to pay the cancellation fees, early termination fees, whatever.
I'm comfortable with taking on the risk of not fulfilling my end of something like a gym contract, provided the mechanism to remove third parties like payment processors.