Hang on..there were 3 coaches not because things were going well but because the place was a one big shit show. That's what Postecoglou inherited. One big shit show. He got no favours from the previous managers ( unlike Rogers at Celtic where Ange left him the heart beat of the team to build on).
As for Conte, if he stayed, Kane wouldn't have, and without Kane, Conte would have been screwed. Last season would have been even worse.
Postecoglou could very well get sacked this year, no great shame in it really.