the people who hold the other keys are the ones most likely to lead a coup against putin.

They are also ones that will absolutely, unavoidably, get informed if Putin wants to launch nuclear weapons. It’s unlikely they will agree, so if Putin ever is to ask them to use their “keys”, he will show himself as an utmost danger to their lives, and they will therefore organize Putin to die within the next 15 minutes or so. And if he manages to hide in a bunker, he will be completely isolated there and will de facto no longer be the POTRF.
Someone else will then take over.