I came to this question by way of the class the government requires of me. We were talking about cover versions one day. The instructor is a musician on the side; he has been a guitar player for 20 years. I mentioned Ike & Tina's version was a better version than The Beatles. He gave no indication of having heard their version. He agreed Tina's vocals had to add something to the song, but denied better musicianship. Now, I grew up with Playboy seeming to always name Paul McCartney the best bass player every year they did the best musicians of the year. I think John Lennon was a great rhythm player. George is a good guitarist and I have seen him in a video clip and admit he was outclassed the more the Sixties went on. I have heard drummers denigrate Ringo in favor of Keith Moon or John Bonham, and I disagree. Ringo does what a great drummer is supposed to do—keep the band together.
I decided the question could be answered by looking at which of the four were asked to play on other's records.
Paul McCartney discography - Wikipedia
George Harrison discography - Wikipedia
With apologies, Mark David Chapman made this entry a bit unfiar. John Lennon discography - Wikipedia
Ringo Starr discography - Wikipedia
Almost no one asked Paul to play bass on their records, Harrison got bored with the music business, I think Lennon collaborating with Elton John and David Bowie needs attention, but The Beatle who people wanted playing on their records was Ringo.
Make of this as you will.
sch 9/22