On a side note, the actual skill that HPMOR is teaching its readers seems to be not science but rather Talmudism, the skill of finding clever interpretations to some words written in a book. It would be cool to see a fanfic that tried to teach science by similar incentives.

This seems to be my impression as well, which is somewhat unfortunate. (This is one of the reasons I try to avoid speculation in my commentaries, though it makes its way in from time to time.)

I think this is the case because there's an author intending the story. The correct way to predict the story is to predict the human writing the story, which is not the correct way to predict the fabric of reality. The story is also (mostly) non-interactive. Identifying an experimental intervention which could have a high information impact, while the backbone of science, is inaccessible to readers. Instead, observational data must be interpreted, which makes it difficult to distinguish between interpretations that agree with the existing evidence but disagree on other features.

And so you could have a game that teaches science (in some ways, Nethack does this, and similar games are easy to imagine), but I don't think you can have a fanfic which does.