I do think its extremely unfair Sage is taking so much blame atm when the support he needed/deserved in the summer transfer window just wasn't there. I'd also question whether someone else could do vastly better with the players he has.
I agree sometimes a change (even for someone of similar or less reputation) can give everyone a lift but I'd question if say Sean Dyche were offered the job, looked at our current defence and players available to him whether he would even be interested in taking over.