The TCI (transistor controlled ignition) box is buried above the front cylinder head, the same place the vboost servo and front coils are. If you remove the left scoop, and the relay plate behind it you should see a black box with a million wires going into it...that's the TCI module. It's a hassle to get out. Since you have a pre-90, you have the older analog box that isn't as reliable as the digitals put in 90 and up. Older ones are known to go bad, where the digital ones rarely ever do.
But from it popping just as you let off makes it sound like bad voltage. My old quad would do that when using the electric starter, but would fire right up if you pulled the recoil. Had a shitty battery. Try cranking it with it on a fast charger, or use jumpers off a car battery (with the cars engine off). See if that helps. If it does, you could have battery issues. Since it starts fine after the initial, it sounds even more like low voltage...it charges up as it's running, but after sitting for a while it doesn't have the oomph. Sounds like a bad battery.
Check the main ground, it's on a bolt near the oil fill. Ensure it's clean.
I doubt oil on the R/R would have anything to do with it. Should be sealed, and it certainly gets wet when it rains. R/Rs do go bad, espicially the old shunt style up to about 97 had. The one on my 97 slowly crapped out. That would result in low voltage with the motor running, possibly low enough so as to not charge at all.
What about bump starting it? I don't think this bike is that tough to bump. Put it in second, and standing next to the bike push it as fast as you can, hop on and drop the clutch. If it starts easy that way....bam....low battery power. Having someone to help makes it even easier.
