I also don't get this "wanted to get Kirk out of their hair" attitude. He was never that much of a pain in the ass and the only big thing he did to piss them off was in Star Trek III. Otherwise, he was a hero of the Federation, over and over. In Star Trek IV they cleared him and gave him back the life he craved with the crew he wanted (I imagine a few of them would have retired if not assigned back on the Enterprise). That's no punishment.
I think a lot of this attitude comes from the DC comics series of the time. They kinda made Kirk something of a thorn in the side of the brass, but the films didn't.
The DC comics were really reflecting the way Kirk was being depicted in the films. But more than anything, it seemed as if whenever there was something big going on, Kirk was involved in it somehow.
The Enterprise A was a dud because it was "funny." Or, because they slapped the ship together so fast they didn't get anything right. Was an old ship? If it was, the controls were more advanced than before, with the touch screens and stuff.
A recently refitted ship would explain that. But then some old-fashioned controls were back in place for the Enterprise
-A in Star Trek VI, go fig.