Does it really pump gas?
I mean did you redirect it and shoot some gas into a jar or something? Because if it really gets gas and spark, and you have good timing and compression, it should run and keep running.
Do the carbs on those things have any modern EPA voodoo electronics? I am suspicious as all get out with small engine carbs with electronic doo dads hooked to them, just seems...unnatural.
Small engines my rule of thumb is it is almost invariably a carb/fuel issue on push mower and smaller engines (well, lawnmowers the flywheel key from hitting crap so they lose time), once they get to riding mower size or larger with all sorts of wiring I think electronic gremlins first, because that has been my experience wrenching on them.
It's like they are designed on purpose to not work if the least little tiniest eeny bitty thing goes wrong, wham, no workee, save the environment! or whatever...save something..save ya from doing your WORK, that's about it...
Heck, I'm a little guy and I can hit tiny bumps with modern riding mowers and it momentarily shuts the engine down just from me bouncing in the seat with the safety switch going on and off. I don't weigh enough even with the settings down low. Buncha crap......how the heck did the hooman bean race even survive without the nanny state for..a zillion generations, is beyond me. According to the goobermint, none of us should even be here, everything in the world was "too unsafe to use" just a generation ago...mumble..rant...