I think that you are selecting only a part of the story. For example, the official boogeyman here is the religion. (By the way, it happens to be associated with political right, at least today in USA.)

Definitely. But there are groups associated with the US political right that are non- or anti-religious. Objectivists are an obvious example. Unsurprisingly, these groups are overrepresented on the internet (though this is becoming less and less the case over the years). My impression is that LW has traditionally skewed toward this branch of the right.

Yet somehow, quotes from Chesterton often get many upvotes in "Rationality Quotes". Does it mean that LW is secretly very sympathetic to religion? Or just that we are able to appreciate a decent quote even from people with whom we disagree on other topics? Could the second explanation possibly apply also to Heartiste or Moldbug?

Yes, but "possible" is a low bar. I do not believe it could entirely, or even in large part explain the frequency of references to Heartiste and Moldbug, or their reception. Chesterton is less famous and less respected than, say, George Orwell, but he is nonetheless a well-known and often quoted political writer in the English speaking world. Heartiste and Moldbug are not. They are so obscure that even having heard of them requires an unusual degree of familiarity with the fringes of the blogosphere.

Your description of political correctness makes it sound a lot like the "Politics is the Mindkiller" gag-rule. The Boogeyman version of political correctness is more like a hybrid of the Cheka and the Inquisition.