I believe McCarthy is not so much claiming that it's "his turn" but that he has earned the right to stand for the position and be taken seriously.
And largely, that is or at least was true.... until he started this round of desperate bargaining with extremists to win the post he covets.
McCarthy did work very hard to help elect Rs to the incoming Congress, including making some low-profile fundraising trips to red states where the base didn't know him or favor him personally but high dollar R donors entertained his pitches and wrote big checks.
The base unknowingly took benefit from those no-spotlight-on-Kevin trips, because the ads that were then funded were indeed for the candidates favored during primaries by that base.
McCarthy didn't whine about views of the base, the common man. He raised money for the election of their chosen primary winners in November because they campaigned under the R banner "for the people" --per their ads-- and electing Rs to the House was the main part of his job, and he did it.
Now some of these rep-elects he helped to win just turn on him? That will be remembered, no matter if they make short term power gains by holding up his election to the Speakership, and by making his professional life impossible in the coming term. They are self-serving ingrates. Pols who are ingrates fall by the wayside because their reputation precedes them amongst their peers and in successive trips to the polls. Their base knows more about them too now. The gloss of their ads and speeches pales next to their antics this week.
McCarthy's hard work and not his waltz with that godforsaken rump caucus of the HFC is why the vast majority of the GOP conference supports him. The question does remain though why McCarthy didn't trust the rest of the conference enough to take a stand and refuse to bargain with the holdouts. The answer is problematic: because he wanted the post enough to put getting it above good of the House, his own party or the country. He's as bad as they are, in a different way and at a different level. How did it happen? Desire for power corrupts even as one accrues more power. It's almost inevitable. But it shows up more often in some than in others. Behold a doomed speaker and a handful of common jerks. One can only hope for the Republic if McCarthy wins and actually manages to tweak the rules and posts demanded by those jerks.