power weight ratio would be critical. You would be lugging a great big piston engine about the sky for the entire life of the airframe.
ability to start and run at high altitude would also be critical. A piston engine's power drops off rather fast at high altitudes. And probably could not run at all above 250 without supercharging, let alone providing any power offtake.
another killer is the separate fuel system required. Maybe piston APUs on piston aircraft made sense, but not on a jet aircraft. The US Navy did fly jets off carriers using avgas for a while, but I don't know with what results. Certainly not commercially sensible.
Thanks for starting the thread. I think we need to ask ourselves fundamental questions like this to get some clear thinking on the subject. Like, why don't we tow aircraft to the takeoff point instead of sitting around with engines running.?