Ah... Of course there's a glitch that prevents progression.
Well, that's not entirely true. It's still possible to progress. It just requires mashing in an area that's not immediately indicative of requiring mashing.
I suspect the game will end up getting hit because of it, which is a bit of a shame. But I managed to identify and fix the bug and things run nicely, now. Until the voting process is over, I can't update the main build. So I apologize sincerely for the inconvenience.
For those who are interested, without going into spoilers, I'll say:
After unlocking a door, you'll enter an area where it seems like you should be able to interact with a Large Thing up a small set of stairs. Please mash Use on it.