Hi and thank you for your question.
This sounds like a classic case of a fuel restriction issue. What it sounds like is happening is that you are running out of fuel in the carburetor and it cannot refill as quickly as it needs to, and eventually the engine quits. This theory is further supported by the statement that the longer you let it sit, the longer it runs again.
IF you have not tried testing with the petcock in both the on and reserve positions, I would start there. There are filters on each pick up inside the tank and one could be clogged. If the ATV has the problem in both the on and reserve positions, you can pull the fuel line off the tank and turn the gas on to see if it looks like it has a healthy flow or not. IF need be, compare the flow rate in the on and reserve positions. They should be comparable and produce a strong stream of fuel.
Beyond that, the most common cause for a fuel restriction is a clogged or kinked vent line. The vent could be either on the carb or fuel tank. I would suggest inspecting the vent hoses coming off the carburetor to make sure they are free flowing. You can disconnect them from the carb and blow through them. They should not offer much more resistance than blowing through your average soft drink straw. Insects have been known to make homes in vent lines, and create the exact symptom you are describing. If you have any doubts about the lines being restricted, simply disconnect them from the carb and see if the symptom disappears.
Typically the fuel tank is vented through the gas cap so the easiest test for that is just remove the gas cap or run it loose. If the cap is the cause of the problem, when you go to loosen it after the engine quit, you might very well hear a whoosh as the atmospheric pressure rushes in to equalize pressure in the tank.
There is a drain screw on the bottom of the carb that you can unscrew once the engine quits to confirm it is a fuel delivery issue as well.
IF you have any more questions, feel free to ask.