Interesting, yes. But I don't get it. He seems to be being blocked by hardcore right-wingers, even though he also is a hardcore right-winger. Is he somehow not "pure" enough, similar to progressives saying (for example) Biden isn't progressive enough?

Somewhat.

McCarthy generally is an "establishment" kind of guy, at least to the eyes of the more extreme members of the Freedom Caucus crowd. As such, he's not willing to use "nuclear option" kinds of tactics to achieve certain goals. There's a subset of the GOP that thinks that they'd achieve more of their goals if they, for example, just let the U.S. crash into the debt ceiling rather than agree to raise it unless they got their wish list satisfied.

So the objectors don't trust McCarthy to go scorched earth the way they want. Some of them are holding out for rules changes that will allow them to force him out at a moment's notice if he doesn't do what they want. Some of them are so distrustful of him that they won't accept him even if he gave them the rules they want. And some of them just dislike him personally because they know he thinks they're awful.

The strategy both sides are employing is to keep ding the same thing over and over, hoping that external forces pressure the other side to fold. McCarthy's hoping enough of the MAGA crowd will get pissed off at the Freedom Caucus and start telling them to cave; the Freedom Caucus is hoping enough McCarthy supporters will get tired of the GOP looking the fool on the nightly news and tell Kevin that it isn't happening.