You know, with Pete, I always felt it was an accumulation of things. I had a band when I was a kid and we were all close friends except for the drummer who lived out of town. And he was OK, nothing great. In the back of our minds we always felt (and had discussed) if someone better came along, we would possibly make a switch.
Well it turns out we tried to make it on the covers circuit and the consensus was we were good, but not great. We had the opportunity of playing with one of the drummers from one of the great bands, who filled in when our drummer was out and it was like night and day. From that point on we knew that our drummer had to go.
I bring this up because it is not unlike the Beatles story. They had such trouble getting ANYONE to play the drums and stick, when they got Pete, he was good enough. But he wasn't tight with them and when you're kids in a band trying to make it, that means a lot. Maybe not so much as adults.
But then they got the chance to play with Ringo, (When Pete couldn't make a gig) and they saw and heard and felt what it could be like with a true seasoned drummer. That was the day, I can guarantee you, the decision was made. The fact that George Martin didn't like him just gave them an out. The die had been cast well before that.
Of course, standard disclaimer that this is my opinion, based on the books and articles I've read about the Beatles.