Allen was a defensive minded coach that wanted to control his quarterback and what kind of plays were called on offense. Sonny had been calling his own plays for years and of course was known for the deep passing game. That was not what Allen wanted to see.
Kilmer was a game manager. He was willing to hand the ball off 25 times a game and take limited chances in the passing game.
Also, Jurgensen pre-dated Allen and had a large following here including Edward Bennett Williams, and that bothered Allen who had a large ego of his own.
Allen traded for Kilmer and as a 33 year old veteran of other teams was never going to eclipse George in popularity with the fans.
I remember someone saying years ago that Sonny being injured in 1972 was not a factor in the Redskins losing the Super Bowl to Miami, because Allen even if Jurgensen had been healthy was not going to let HIM be a hero and win the game for the organization with a late comeback.
He simply would not have benched Kilmer in that Miami game, even though if available Jurgensen might have been able to rally Washington to a win.