I'm not familiar with the FS80 and that specific carb, but I believe the primer is the old style (pushes fuel to the carb), versus the new style that pulls fuel through the carb. That changes the possibilities a bit, but it shouldn't really matter.
When you say it runs until the primed gas runs out, I interpret that to mean you press the primer bulb a few times, start it, it runs for 5-10 seconds and quits. Repeat and it does it again. Try this test - with it running, keep pressing the primer bulb. Does it keep running longer (until you stop pumping)?
I expect it will which narrows the problem down to the fuel pump portion of the carb not working. There are 2 typical modes of failure. The diaphragm in the carb could be stuck or stiff and is not able to move to pump fuel, or, the carb is not getting an impulse feed from the engine to cause the diaphragm to move.
I'd check the impulse feed first. There will be an impulse line or passage from the engine to the carb that provides the pulse needed. For carbs with an external line, many times they crack or fall off or just don't seal. Confirm that it is present and not cracked or blocked. Better still - replace with new. (If the feed is an internal passage between the engine and carb body, remove the carb and check it isn't blocked by feeling for air pulsing when pulling the engine over).
If the impulse is fine, then you need to open the carb back up and confirm the diaphragm is properly installed, free to move, and not stiff or cracked. And that the impulse passage within the carb is not blocked.
Beyond that, I'd be looking at the integrity of the fuel line and checking the fuel filter/pickup and screens / passages in the carb are not gummed up with stale gas or goo. With a pressure primer, a cracked fuel line will still deliver gas to the carb under pressure (when pumped) but not under vacuum (running) and a pressure primer could easily overcome a blocked screen that the carb pump can't.
