Part of the problems with it is likely long term usage because wood is not an entirely predictable fuel. All sorts of hydrocarbon oils and tars can come out of it and the moisture content of wood can be all over the place.

Possibly modern wood pellets would eliminate many of these problems, but if you aren't getting a really good burn, which takes some skill to setup with just random chopped wood pieces, you may end up gunking the engine all up and filling the oil with crap and possibly having some not so great exhaust coming out.

Otherwise you need the skills and an engine simple enough to be worth semi regularly opening it up to clean all the carbon and crap out of it. Something that might not seem like too big of a deal when people already use 1930s cars, but would become a much bigger and bigger deal in the decades after WWII when cars and engines become increasingly complex and people don't expect to be removing major engine components after 5,000 miles.