It was an Alpha move. But it was Alpha from the standpoint of Paul McCartney wanting to dominate the band -- and the world -- musically. John might have been the leader of the Beatles on a social level and in the media, since he was the best speaker. But Paul did the musical planning and arranging, as has been documented in countless books.
Best's drumming was so amateurish it was embarrassing. They were mostly using him because his mother, Mona, owned a club where they played. When they finally got a record deal after years of struggle (they were rejected literally everywhere) the producer said he would not use Best on the records.
That producer turned out to be George Martin, who shepherded the Beatles through some of the best music of the 20th century. After a rocky start, Martin came to rely on Ringo as the most dependable Beatle. In Mark Lewisohn's "Beatles Recording Sessions" book, he states that after listening to thousands of hours of Beatles takes, it was Ringo who most rarely caused a take to break down because of a mistake.
All that said, there was personal/sexual jealousy in the Beatles ranks but it was mostly directed from John to Paul, since Paul snared all the babes and got all the attention when he stepped on stage. It's doubtful Paul would have been too jealous of Pete Best because Paul apparently had more women than he could handle -- even before the Beatles were famous.
The scenario in the Paul vs. Pete battle is that Paul was a musical perfectionist. Best couldn't cut it, and that drove Paul crazy. As the Beatles progressed, George would start to drive Paul similarly crazy and starting with the "Help" album, Paul started playing lead guitar ("Ticket to Ride," "Another Girl"). Paul also played every lead guitar solo on the "Sgt. Pepper" album save one: the lead on "Fixing a Hole." Paul's musical domination of the band eventually drove even the laid back Ringo to quit temporarily in 1968 for being told how and what to play. More than Yoko it was a factor in their breakup. You can see the others wince in disgust when Paul starts giving musical direction in the "Let it Be" movie, after which George shoots back some sarcastic comments at him. A few days later George would quit. Heck, even the musicians in Wings bailed, telling the press they felt like they were turning into "McCartney-programmed robots."
For more details on Paul's musical Alpha-ness, read recording engineer Geoff Emerick's book "Here, There and Everywhere."
For an example of how Best could not hold a steady or authoritative beat, go here:
http://youtu.be/OwzY0XYVyTk
Or listen to how he speeds up during the solo here:
http://youtu.be/BPfzQWpQCpM
Or check out the Decca audition tape. They didn't pass the audition.