Glad it's not the compression. So, where are you at with the saw? I'm guessing your problem is still not resolved? Sounds like a fuel issue, however, how is the spark? When having trouble with it running, remove spark plug and ground it to the head, give it a couple pulls and look for a nice blue spark, if thats OK, then pretty safe to move on to fuel related problem. But, to be safe, check themagneto and flywheel. They should be clean. If the flywheel has rust on the magnetic portion, clean it with fine grit sandpaper and brake clean. Same with magneto, make sure it is clean of oil and dirt (at the tips near the flywheel).
Probably could stand a rebuild of the carb. A kit is about 5 bucks, rebuild manuals can be downloaded from the internet. Give it a once over with lots of carb cleaner.
Your tank vent has me thinking. Is your saw designed to vent thru the cap or someplace else? I'm thinking someplace else. Need to locate the correct vent and see if it is clear. When was the fuel filter changed? Check the impulse line for a leak? Maybe fuel line has a leak and when tank drops to a certian level it exposes a hole in the fuel line? (just guesing at this point)
When the saw stalls out, remove the fuel cap, listen for any vacume. Try to restart and see if it runs now. That will tell you if it is a venting issue. Same thing with spark, check it after it stalls, sometimes coils or plugs can get flakey after they heat up. Another check would be after it stalls, prime it, then see if it runs.
Other thn that, look for carb or impulse line leaks by spraying wd-40 around the base of the carb, and the impulse line, if the way the saw runs changes when spraying, you found a leak.
Based on info provided, I'm thinking fuel venting problem, or carb needs a once over. Good luck, KD