I dont see what could be burning. That compartment is made for flame.
Why not just leave the pilot burn for several hours and leave the
burner turned off. That will dry things out. Or run a fan into the
burner compartment for a day first. Once things dry, it should work.
Look for something that may have floated in there such as a piece of
wood. I cant see any reason for the burning smell.

Well, since the last time I tried lighting the pilot, it exploded on me
(not badly enough to escape the chamber, but there was certainly some
unexplained excess gas in there after having the gas shut off for over
10 minutes). I don't feel lucky enough to keep trying to light it,
especially since it apparently "popped" once on its own before I started
messing with it.

Thanks for the ideas, I do plan to do a "postmortem" on the old 30
gallon once the new 40 gallon is installed. I was going to go for a
tankless, but the extra effort of installing it, extra up-front cost,
and the fact that nobody seems to keep them in stock around here pushed
me back to installing a short-style 40 gallon and putting it up on a
support off the floor.