The barrel can make a difference, bore size, surface quality etc all can have an effect on the way the ball spins. I'm just not convinced that the different air cushions from poppet/spool (i.e lower volume and higher pressure for popper, higher volume, lower pressure for spool) could make such a difference when the ball still has to travel through a 14" barrel.
The way to test it would be with the same barrel, same paint, marker set up in a fixed stand, ideally bolted down! with a fixed camera behind it, firing over a set distance of say 30m into a sheet with a grid on it. Fire 30 shots through each gun. From that you could analyse the footage and see how the balls fly, from that you could determine if there is a difference in their flight. But ideally you'd have to do it with a few different makes of spooly and a few different makes of poppet, all using the same barrel and paint.
Wouldn't do it myself because I haven't got the time or equipment to do it, but that's a way it could be tested.
I may well be wrong, and am quite happy to be proven wrong. But IMO the biggest factor is paint and barrel, even if the air cushion did spin the ball, which logically is unlikely, it has to travel through a straight tube which would even it out, I think...