Not sure they are comparable, the Chelsea Dr was a well liked member of staff and most will have thought it was an unjustified way to treat someone.
Get the impression that due to Barton's general attitude he will not have been popular in the dressing room and the players may have been happy to see the back of him (results did briefly pick up when he was out the team).
Some of the players will lose confidence in him if he continues to stick with 4-3-3 and the results stay the same.