I had always heard it was Fay Vincent who held the grudge against Rose and he lobbied the Hall of Fame to change its rules.
That said, does it really matter anymore? If Pete needs the Hall of Fame recognition to be complete, he's likely never going to fill that hole. The Hall of Fame still stands with or without Pete. Pete's accomplishments are mentioned throughout the museum, so it's not like they've tried to whitewash over his career.
Regardless the equating of betting to PED's is completely useless. You can argue back and forth over which one is worse for the game, which one is more damaging to one's integrity. But one is not the other, and there's no reason why the penalty for one has to equal the penalty for another.