Hekllo JasonB
You have lot's of good input so far.
I apologize in advance if these questions seem dumb because I realize you already know a bit about bikes. 500’s are ”bleepin” finicky. You can try the following.
Do you leave the gas on after riding? The float might be high or the needle/seat might weep slowly filling engine after a while and subsequently flooding it.
INSPECT CARB BOWL GAS - Drain float bowl as dave8338 suggested but drain it into a narrow clear glass. What color is the gas, any sediment, is there a small clear bubble in the bottom of the glass? You can get sediment and water from the gas at the station.
GAS LEVEL TEST – Take a clear 10” plastic tube, connect it to the bottom of bowl drain. Hold it next to carb with open end even with carb top. Open drain valve. Gas level should be from even with bottom edge of carb body to 1/8” below it.
PRELIMINARY TIMING CHECK – Even it does not run you can hook a timing lite to it check timing. Mark the flywheel timing mark and the timing mark on eng case with white out. This makes marks highly visible. Remove spark plug and ground it somewhere. Some victim kicks it over while some lucky person gets to hold the button down on the lite and check the timing. Do this in a dimly lit area. Obviously if it’s on the mark your flywheel did not spin on the crank so that’s ok but retighten nut anyway.
COIL AND STATOR CHECK – Ohm coil and stator per manual. See manual link below. If they ohm in range then they are 99% going to be good. If it’s the 500 cc stator then the stator spec should be near the same.
http://www.kb0nly.info/ATC/index.php?dir=ATC250R%2F
STATOR VOLTAGE OUTPUT – Put volt meter on 50, disconnect stator wire, connect red lead to stator and ground the black lead to eng. If stator has 2 wires connect one lead to one wire and the other lead to the other. Voltage should be 30 – 50 while kicking. A digital volt meter that has a peak reading setting is best otherwise it may or may not work. Watch meter while kicking if it does not have this feature.
SPARK PLUG COLOR – It should be a strong pale to bright blue as it is now on that set up. Check it AFTER trying to start it.
PLUG WET OR DRY - Remove after kicking 4 times and see if it is bone dry, damp or dripping wet. Bone dry speaks for itself.
IGNITION
PARTS – What exactly is the CDI, Coil and stator off of?
HOT COIL – Try your other bike and see how hot it gets then compare it to your other bike[unless this is your other bike, lol]. It MAY be remotely possible the stator in the 500 eng is putting out too much voltage for your system therefore causing you coil to get hot. I actually don’t know if this is possible with your particular system, perhaps someone else will. Guessing won’t help.
COMPRESSION RELEASE - Does your compression release take a cable or is it designed to be manually operated? If it’s factory or your compression is higher than stock then it MUST work properly to start this pig. If it is closing while you are kicking it, it will likely never start.
COMPRESSION TEST – Do one if you want as lastfooler in VA suggested, but it has no effect on your problem [providing your compression release is working properly] since it ran fine once already but it won’t hurt to know what it is. You might need race gas to reduce detonation potential if you don’t use it already.