86 Fourtrax 350 starts and runs but shuts itself off

I like a lot of the help that others have got here but I did not see a match for the issue I am having so here is my story.

I got my 1986 Honda Fourtrax 350 and ran it on a number of trips for a year with no problems until its last trip where I left camp and made it a short distance out of camp before the machine just shut off and would not start back up as if there was no power. After towing it back to camp and letting it sit for hours it started right up with no problems and shut off again after running a short period. What I have found is it will start up fine and runs for between 10 and 12 minutes shuts off and will not start back up. If you come back hours later it starts right up. Something else I’ve noticed is the oil temp light next to the key is supposed to come on briefly when the key is turned and then goes off but while this problem has existed sometimes the light does not come on but other times it does but no matter what the light is doing the shutting off after 10 minutes still happens.

After trying to understand what this problem could be, I am assuming it is an issue with the oil temp sensor or fan controller or cdi. If you agree, is there any way to test these components to know which is failing? If not do you have a feel for which order to start replacing? Also if you think there is something else causing this problem I would like to hear about it. Thanks!