Couple of other ideas: Check your valve lash. That is, the rocker arm clearances for both the intake and exhaust valves. While valve lash usually INCREASES with wear/running, sometimes people will mess with it and adjust one valve too tight.
When this happens, the engine will still have some compression, but the combustion chamber won't seal properly and it'll be Hell to get started. Most 4-pokes like yours run with an intake valve clearance of .004", and an exhaust valve clearance of .005-.006". Set it with a feeler gauge. Be sure that the valve you are adjusting is fully closed when setting the clearance.
Valve lash OK, or too loose? Not the problem, then. You might also want to temporarily unbolt the exhaust header and try the starter again. If the quad has been sitting for some time, the silencer or header pipe may be plugged up. With what, you ask?

I've blasted mouse nests, bird nests, dirt daubers, and dead rat carcasses out of mufflers during my time as a tech. If there's a foreign object in the exhaust, you can kick it all day long yet get nowhere....