In the OPs case I'm almost willing to bet money that it's an ignitor or similar ignition issue because it's an instantaneous switch from running great to not running at all, then back again. Your issue sounds a bit more like dirty carbs or air leak issues, which are very common on these older bikes. Having to push start sounds electrical, but it's not really clear from your description if the starter just isn't running, or if you've tried so much to start it that you've run the battery down.
In any case, I see this is your first post so would recommend that you post a "hello" in the GS Owners forum. Add your bike to your sig, your location in the user CP, and post a pic of your bike - that will impress
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After you've done that, come back and start your own post about the issues you're having. Make sure you give a full history of your bike as you know it to date. Then you can get some individualized attention for dealing with your bike's issues.
Oh, and prime is meant to allow fuel flow without engine vacuum: but that fuel flow is less than on "on" or "res". As named, it's more for priming the carb float bowls when they might be empty, and is best used in 10-20 second bursts when the bike has been sitting for a few days/weeks. You don't want to go riding like that. Common petcock issues are the petcock allowing fuel without vacuum in the "on" position and fuel leaking through the diaphragm and vacuum hose, which can flood one of the cylinders (#2 in your case). You can easily check for both by disconnecting your hoses and having a look.
Some questions you'll be asked:
Have you adjusted the valves?
Have you cleaned the carbs (no, REALLY cleaned - disassembled, overnight soak in dip, and re-assembled with new o-rings)?
What shape are your intake boots in?
Have you changed your intake boot o-rings?
Is your airbox completely/properly sealed? (are you running the stock airbox or pods - if the latter are you sure the bike is properly jetted?)
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