Just been thinking of what I would do to a saw if it had these symptoms.
The poor idle - I would wind the idle in until it would either fast idle or not even run with the screw turned fully in.
If it won't idle with the idle screw turned fully in, I would see what effect adjusting the "L" screw has, typically on half a turn out, the engine will race as the mixture will be a bit lean!
Sometimes the L screw needs to come out more than one turn, have you tried 1 1/2 turn, 2 turns, 2 1/2 turns?
The poor starting and weak idle could be an over weak low speed mix!
Have you checked the spark when the engine wont start, can you try this with a mild speed start to see if the spark is a bit iffy at low crank speed?
Other than that, it sounds much like lack of compression but know you have tried this with a gauge - if all else fails, drop out the base gasket and seal with a little liquid gasket and see if that gets the motor working, if it does, it looks like the piston is just worn out. Have you checked the ends of the top ring and compared their depth to the depth half way round - they can be thin on high hour engines. I had an old hedge trimmer with the same fault so did this quick fix as it was just for my use and gets 1-2 hours use a year!
The last thing to check is for carbon in the exhaust port and also for the exhaust being blocked with carbon and crap - this can cause lots of oily deposits around the exhaust - has it got a spark arrestor that has got blocked?
Thats about it - hope you rebuilt the carb with the pump membrane gasket closest to the carb body and the spacer between the carb and the diaphragm.
Oh - also check the fuel line and filter are in good nick
Carbs can be un-fixable, I had an 020 carb recently, just wouldn't idle consistantly so must have had a low speed issue but nothing fixed it - can you try a known good carb?
If it happens again, try running it with the fuel cap off - I know the vent has been changed but worth a punt!