In the sense of a farmer being more likely to have access to a local supply of firewood, and that tractors are probably more used for longer stretches at a time than running down to the grocery store, sure.

Historically, they weren't that common, as large-scale use of wood gas was mostly a thing in Europe during WWII, and during that period continental European agriculture was still mostly horse-driven. After WWII when agricultural mechanization really picked up, fuel was again available so there was no big motivation to put up with the disadvantages of wood gasifiers.

I think it would, the only problem being smaller row crop farmers who would be mostly likely benefit to implement it or want to implement it have been pushed out of agriculture more and more over the decades and struggle to survive at all. Which makes spending time and money on experimental work like this far less likely.

Check out http://www.driveonwood.com to see plenty of examples of both. A wood car or truck can be amazingly practical for any use involving long steady state (i.e. highway driving), not so much for city use.

A tractor can certainly work well on wood gas.