Several things to check.

First, look at all the fuel system hoses for any cracking/splits/etc.

Check the carburetor mounting studs/bolts and gasket for looseness or damage.

Then, remove the muffler and disassemble it to check for carbon build up on the screen. While the muffler is off, inspect the cylinder wall and piston. There should be no grooves/lines in either the cylinder wall or piston. Any grooves means the head/piston are damaged and not repairable.

The cost of a new cylinder head and piston and labor are usually within $50 or so of a new saw---so it makes better sense to buy new than repair old.

The symptom has fuel filter wrote all over it. So much so that were you to give me the saw rather than spend additional time and money,I would install another fuel filter.
Here's why I'm convinced it's the filter. Time it takes to cool alows fuel to slowly soak through and fill the filter. The contents of filter plus the small amount that is able to soak in is about what it takes to run the saw a few minutes before starving out. Don't use a high quility brass filter because water can not pass and will block fuel. I also suggest a new batch of fuel from a high volume station in a clean container. Gas curently contains various amounts of alcohol. Alcohol atracts and holds moisture which in turn plays all kinds of havoc with fuel systems,small and 2 cycle especialy. Last but certainly not least,treat ALL FUEL THAT WILL NOT BE USED WITHIN 2 WEEKS with stabilizer when you buy it.
That saw sounds like a match for me. Work 5 minutes then tell swmbo I must alow saw to rest and cool so I may as well have a beer while i wait.

Fuel line or the fuel tank is not venting.