First of all, the charge does not simply go away because you sign an affidavit... there is a little bit more process than that -- you may need to invest a couple of hours into it. At the end of that the charge will, in almost all cases, be reversed.

But that only makes ONE charge go away. The merchant can then submit another. And another. Every. Single. Month. Think of it as a DOS attack. Closing the card SHOULD (but apparently doesn't?) prevent that.