It may just be as simple as "we dont think you are the right man for the job anymore", and then him saying "fine, to hell w you then".
All the while he still hasnt gotten past the 2nd round. I'm not one to beat that drum hard, as I like Trotz...but it's a drum that is there, and it will get beaten.
It may be, but that's assuming that it's the organization who is willing to move on.
My suggestion is that I haven't seen anyone indicate that maybe Trotz just doesn't want to come back. Maybe something about working here rubs him the wrong way and he no longer wishes to coach the Capitals.
IF that's the case, what would his perspective look like? Why would he choose to leave a team with arguably as much of a chance as anyone else of doing damage? Is it the players, or is it management? Honestly, if it's the former it's probably also the latter, since otherwise you'd just work with management to move the players you don't feel are fitting in, but either way, what would Trotz's opinion be at its arguable "worst"?
That's the mindset of the original post, and of course it was all speculative. It was trying to look at the situation from another angle, and then re-evaluate the motives behind everyone doing what they're doing and see if they make any sense.