Oh, dear. Stop thinking of the Beatles as just a band. They were a freaking phenomenon beyond the music.
But first, let's start with the music. Ringo was the only drummer the Beatles could ever have wanted. Period. Listen to their earliest records. Ringo drove the band. Besides the obvious hits like "She Loves You" and "I Want To Hold Your Hand", listen to him play on "Long Tall Sally". Who else was available in 1962? Would you have kept Pete Best? Who else? Tommy Moore? Johnny Hutchinson? Andy White? Or maybe Jimmy Nicol?
On to personalities. John has been quoted as saying, "Pete Best was a drummer; Ringo was a Beatle." Ringo fit right in with the others, Liverpool humor and all. As Sir George predicted, the Beatles would be known as much for their personalities as their music. Those of you not around in their heyday have no clue about the effect of the Beatles' personalities on their success.
Ringo was subordinate and compliant. He was a team player. He played what he was told to play. John's first instructions to Ringo upon his joining the band: Comb your hair forward, shave the beard, but you can keep the sidies (sideburns). Can't say Pete Best ever did that. Ringo had little in the way of big ego. He deferred to John and Paul without acting like a prima donna.
Then there's the movies. Ringo was the focus of both AHDN and Help. He stole the show in both cases. You have to appreciate how both films kept Beatlemania alive in '64 and '65. Who but "our poor, little Richard" could have played the hapless fellow in those movies? Pete Best? (Laughing up my sleeve - "Ho, Ho!")
At the height of Beatlemania, Ringo's fan mail outweighed the others. He was a fan favorite in a way far different from the others.
Ringo did some wonderful fills on records from their middle period. Even some of the best drummers can't quite reproduce them, including the great Hal Blaine. Imitate, yes. Reproduce, no.
You've got to understand, the Beatles' success was based on chemistry among the members, of which the music was only one ingredient. In his own way, Ringo was an equal to the others.
As always, in the words of Dr. Winston O' Boogie: "You should've been there."