Couple of hints.
I have a Briggs and Stratton spark tester # 19368 (about $15 from flea bay) that can readily be connected in series (engine will run while it's connected in series if it has spark) with a spark plug wire to watch for intermittent ignition spark or you can clip one end to ground and check spark while you crank the engine. (it has a clear viewing window) You can see the intensity of the spark and if it's weak it's readily apparent, but to view a really hot white spark the little window may have to be shaded. I use this Briggs type instead of the spark type that has the light bulb because the lamp type will still flash with weak spark.
The other option for you to check for spark (to get an idea if it's CDI ignition failure or ignition spark failure) is too keep a spark plug readily available and connect it too the spark plug wire and get a good ground on the spark plug body.
You indicate that you can let it cool off and get a re-start and when it quits it immediately quits like you used the kill switch. If you find you are losing ignition spark investigate the kill switch also.
Next you indicate that the cooling fan runs sometimes when the engine quits and in cold weather in the snow and the engine might be overheating.
Here is a link to a IR thermometer ($18) with free shipping. You can use one of these to actually get a instant temp reading of you block.
I think the oil temp sensor that activates the fan is listed in the service manual. (the temperature at which the oil block sensor activates the fan and the red temp light)
I'm not sure but I seem to think the block temp or oil might be around 180-190F but not sure. (anyway the NON-contact IR sensor will tell you immediately the block temp) I use a IR thermometer for several things when working on engines and heat air. Goober and or Shadetree may know the temp of the oil sensor. See that the oil coolers fins are not clogged with mud/gunk.
Next: If you find out that it's fuel delivery issue, their is a fuel pump control relay up under the right rear fender under a cover that controls the fuel pump. If this relay gets erratic the fuel pump will get erratic, but usually a choke would be required or a longer crank time to re-start the machine. (get gas to a dry carb)
Your description of the beast kinda indicates ignition shutdown. (instant failure)
When the cooling fan runs does the oil temp light come on?
(if the RED temp light comes on I would quit running the machine until I checked the block temp due to engine being damaged)
If you don't have a 86 Fourtrax service manual you can download one from this site.
Keep in mind that a bad battery cable connection will shut down a 86 fourtrax ignition because the 12v is feeding the CDI module. (must have 12v bat to run the ignition)
and IMHO you need to be concerned about the cooling fan running in cold weather. (not a good thing)
Section 20 of the service manual has some real good troubleshooting tips for the Fan control.