Oh please. She's using her column to describe her perception of her little part of the Reagan White House, including the office politics and personalities involved. She settled an old score or two, but not in a very meanspirited way. She did what people do. She's human.
She might have been better off leaving the score settling for another column, because this column was absolutely for the ages prior that point, but, for reasons best known to her, she decided to include it. She has an absolute first amendment right to include whatever she wants in her work. If her bosses don't like it they can yank it, but, as we can see, this stuff must've passed muster because it's in.
No she doesn't. as you note, her bosses can pull whatever they want - - she's their employee and doesn't have any "right" to have ANYTHING published, let alone a small, mean-spirited snipe at some old personal foe, which ends up marring what was supposed to be a homage to Reagan.
I will grant you this -- the column was quite long. Some of it could easily have been trimmed. The part about the other speechwriters remind me of things I have written in posts here that when I'm reviewing it decide that they are better off left on the cutting room floor. Still, that she decided to leave it means she cared about it very deeply.
The column was titled The Ben Elliot Story, so this was central to the theme. I think that since she was writing about the death by a thousand cuts suffered by this guy she considered a mentor, smacking back at the hacks in the Reagan WH also must have been central. Perhaps not for her, but for Ben.