That's what I do once in a while... it works.

Yeah but such a horrible solution. After the pain of reporting it and getting it changed, now you get to update a dozen or more auto-pay accounts. You're sure to miss at least one and have something (probably important like your internet or phone bill) cut off on you at the worst possible time. Plus you get to spend days without usable plastic. Hope you have a spare card or cash.

Would much prefer a solution simpler/easier/less devstating than dropping Mjölnir on it.

It's not a guarantee - card networks have a system where they will send payment processors the new card details, so the fraudulent charges will recur.

As far as I know, it's not granular, per-merchant, so you can't skip the fraudulent merchant while updating the legitimate ones.

Furthermore, some banks may not even give control to the user about whether this happens or not.