I'm not sure I understand this. Yes, the hand-picked successor will be able to fulfill the rest of the term, but then he's up for re-election. If people were actually outraged about this, they could vote him out the next time there's an election (just like they could have voted out the predecessor that set this process in motion). If people were really upset about corruption, they could have voted Cuomo out for killing the anti-corruption panel when it started to look in to corruption connected to him (IIRC, this happened just a few months before the 2014 primary).
It's hard for me to believe that these things really bother people when they keep voting back in the people who are doing them.