That's not how the burden of proof works. In a credit card dispute the merchant needs to have there be no ambiguity that you did, in fact, authorize the transfer of money to their account and that the authorization is still in effect. The absence of evidence means you win.

Typically, the way it would work is that they would say, "You authorized it," and you would say, "Actually, I didn't authorize that payment. I only authorized the prior payments. I never authorized the subsequent ones because I logged in to their site and attempted to cancel, but they did not provide me with a means to do so."

To which they could say no your initial authorization should also apply to these subsequent charges because that cancellation method is not one we accept. In the past, that defense could work. Your NEW counterargument would be: Sorry, but they cannot rely on and profit from unlawful behavior.