If you ask a catholic priest whether the position of the church on abortion is important for him, I think he will say "yes".

It's not an exact analogy, but it's close because it's much more important to outsiders than to insiders.

The basilisk isn't directly a LW idea. but the basilisk follows from LW-style ideas and is close enough that Eliezer couldn't just say "nothing like the basilisk could possibly work". A closer analogy may be more like, oh, geocentricism. The church does not believe that geocentricism is true any more than LW believes in the basilisk. On the other hand, the man who became the current Pope has pretty much said that the church was right in its treatment of Galileo even if the church was wrong about geocentricism itself. And you still see this used to criticize the church. And I doubt that many priests would think that the way the church treated Galileo is very important compared to either abortion or Jesus dying for our sins.