You're thinking like an engineer. From the perspective of the bank, you (the individual) has given a merchant (gym etc) a standing authority to bill you. That authority is between you and the merchant, not between your specific card number and their payment processor. Having your card reissued is not a withdrawal of that authority.

Based on the brief detail provided, I think OP's mistake was to cancel/reissue the card rather than dispute the charge / block the merchant.