I'll bite, I guess.
If you look at the early songs that sort of defined the Beatles and that most fans would associate with Beatle popularity, they're overwhelmingly Lennon cuts (Please Please Me, Help!, Ticket to Ride, I Want to Hold Your Hand, Twist and Shout, A Hard Day's Night, I Feel Fine) with only a few exceptions. Paul was always a great songwriter, but it wasn't until Sgt. Pepper, I feel, that Paul emerged as a songwriting talent near or equal to the level of Lennon. Then, after Brian Epstein died, it all started to unravel. Paul felt it was his duty to "rally the troops" and made himself the unofficial Beatles leader. Lennon got tired of it, as did Harrison. As Lennon said in an interview, "We all got fed up with being side-men for Paul." Paul became more and more picky with his music and less open for input from his bandmates. If you look at Paul during the Let It Be sessions, he looks like an arrogant ass, and it all sort of comes to a climax with his arguing with George Harrison during the infamous scene about the guitar part of "I've Got A Feeling."
(If you want my opinion, I've Got A Feeling is George Harrison at his finest. Listen to the song, every second of it, focusing only on George's guitar. I truly believe it is one of the finest examples of great guitar work on any song. Any time I hear it, I want to give Paul the middle finger and say, "Stick it up your arse, Paul. George's work here is genius.")
Paul wanted so desperately to be the true leader of the Beatles. Perhaps better put, he wanted to be the true leader of ANY band. After Paul left the Beatles, he got his wish. He started a band where he was the headliner and the rest of the band members were only sidemen.
(As an aside, besides diehard McCartney fans, how many people can name any member of Wings besides Paul? I know I can't. But, funny enough, when Wings plays in concerts, the songs they play are overwhelmingly Beatles songs. That's interesting when you think that Paul was with the Beatles for only about 10 years and has been with Wings for nearly 40.)
So the beef we have with Paul, in a nutshell, is that once he became a great songwriter he tried to turn the rest of the Beatles into Wings. Though John is also guilty of this, I feel that he also stifled George to the point where it became unbearable for him to be a member of the band. His role in the Beatles was and always should have been a notch below John's, and once Paul tried to change that, the Beatles unraveled.
I should add that the above post is just my opinion, and I know there are glaring inaccuracies and biases in what I have just written. I just tried to sum up the "Anti-Paul group's" perception of McCartney as best as I could. If someone were to critique Lennon, they could write pages and pages about him and his relationship with Yoko and how detrimental it all was to the Beatles, and there would be little room for argument. It all depends on which band member you like more and whose side you tend to fall on. I am and always will be biased toward John and George.