Maybe not your exact same problem but I have a older (metal tag) 036 Pro that would start right up and run, rev and cut but wouldn't idle for more than maybe 5 seconds- tops, was very erratic and had a terrible hesitation when you squeezed the throttle. Seemed like it just needed the low side carb screw opened up, which I did, and it helped, but still wasn't right. Carb had limit caps so I removed those to allow me to open the low side up some more. Saw got better but still was not right. I rebuilt the carb. Same Issue. I decided to just go with a new OE carb. I went ahead and removed the limit caps before installing it. I also replace the fuel and impulse hoses while I was there.
Approximately $100 and however long it took me, lo and behold-same issue. Funny thing is this saw would start just as easy as any perfectly good running saw which was steering me away from it having an air leak. So after a bunch of scratching my head I decided it was time to dig a little deeper-
So I take the saw apart and do a visual on the piston/cylinder/rings, check ring end gap, all of the normal stuff. I check the armature air gap, intake manifold, crank seals, plug wire and pretty much everything I can think of that could be causing this idle/hesitation problem. I am finding nothing obvious.
By now it's time to visit the beer fridge. As I am standing there enjoying my refreshing frosty beverage I just happen to notice something quite odd. On the clutch side of the crankcase where the case is machined for the cylinder base gasket there is a piece of the base gasket approximately 1/4" long dead center in between the front and rear cylinder mounting bolts that is non existent. I probably would have noticed it sooner but it wasn't like it was torn or upset in any way it was just as if it progressively went away over time. Needless to say-That was the problem. New base gasket, cleaned up the cylinder and fitted a new set of rings since I was there and even put the old carb back on. That was over a year ago. Saw hasn't had one single issue since. I just had to tell this one because it is probably as UN-COMMON as they come. You just never know what you will find.