That is exactly why I call those a bug screen.
Cause thats about all its gonna stop from going into your engine.
Leave the starter cup alone. Find a new recoil or fix the one you have.
Your rope works better cause your getting more revolutions per yank than the recoil was.
A good rule is dont go poking your fingers where you wouldn't poke your di... um, you know..
Sitting dormant is the worst for engines.
The gas goes bad and turns to varnish as you found out. Just cleaning the jets might get most of the crud out but there are little air passages that must be clean to function properly. This is the main reason people have issues.
If it only runs with the choke on or partially on, the carb is plugged and not getting enough fuel. Restricting the air with the choke fixes the air/fuel mixture but its not the proper way to run things.
In the future, if you know its gonna sit for a while, drain the gas and let the engine run till it empties the bowl and sputters out on its own.
