Yes and it didn't happen quite like that according to ML, it was George who instigated Ringo joining, he asked Ringo to join even before he went to Skegness, at this point Ringo was only a kind of stand in drummer with RSATH having already left them to do the gig in France.
I like how he analysed it thus;
John brought Paul in, Paul brought George in, George brought Ringo in.
What seems to be rubbish though is the theory that Pete was sacked for money reasons, the reason he was sacked eventually was that he never socialised with J.P & G and never really spoke to them much, mostly though his drumming was poor, his timing was out and he didn't have much of a range of beats that The Beatles needed, when he was ill for a couple of bookings Ringo stood in and they noticed a huge difference in the tightness of the band.
The final straw came though when George Martin told them he would always have to use a session drummer on their tracks, they had this vision of a non Beatle playing with them, someone they didn't know....maybe even some middle aged guy or different guys playing on ALL theyre recordings into the future, they certainly didn't want that, so George Martin more or less confirmed what they already knew - Pete wasn't good enough and couldn't keep time. Pete was a goner from that moment.
It seems Pete's poor drumming in recordings had been noted when they did the Sheridan recordings as well. So Martin's view wasn't an outlier. Sloppiness that you can get away with live is shown up pretty fast on tape.
The Lewisohn book doesn't make exactly clear what Ringo did to get substituted on his first attempt to record. It suggests he tried to do too much including using a tambourine. So it could have been a bit of a brain snap rather than poor drumming per se. The important thing is that Ringo clearly drummed well enough when given the next chance the Martin didnt bother re engaging White, even as a stand by.