Presumably, because most (all?) stirling engines are not self-starting?
Also, far cheaper to manufacture I'd guess.
I too have plans to build a wood-burning stove-top stirling engine powered fan - but I only have 8" of vertical height to play with (so no commercial fan will fit).
I always wanted to build something a bit like this:
Without the tealights, and with a fan on the front, I reckon that'd only take up about 8", stopping it safely might be quite interesting though, and it would have to be very low maintenance (self-oiling or greased-for-life somehow), as it would be my old Mum's stove it lived on top of.