Finishing up.
When the saw was running, the vibration was hard to miss. My teeth were chattering, and my fingers went numb within a minute. The flywheel has two broken fins, and it crossed my mind this might be a contributing factor. I pulled the starter off, removed the flywheel, and used an angle grinder to remove the remains of the broken fins. I then ground away two more fins, each on the opposing side of the already absent fins. I thought this would help with balance. I then re-installed.
When I restarted the saw, it was running even more rich than before? Lots of smoke, a sopping wet muffler, and after I shut it down, it wouldn't restart.
...but I never touched the carb!!! Well, except for the three times I stripped, cleaned, and reassembled it. The last thing I touched was the flywheel, and I noticed when I remounted it that the ignition coil was just brushing the magnet enough to leave a scratch on the face. Can this affect performance? I reset the coil using the business card trick as before. That helped a lot! It started about the same as before removing the flywheel, and after a little tuning, just sort of idles along without complaining, racing, stalling. It even seems to accelerate well. I held a blank sheet of paper in front of the muffler while it was idling, and no spooge appeared on the sheet. More importantly, the vibration was/is significantly reduced. Since the flywheel is really used to push air through the cylinder fins, I hope this doesn't reduce air flow too much.
Convinced the carb was OK, the next problem was the fuel tank. It's not sealing, and fuel is running out past the cap. This is not a little fuel, but enough to impact mileage and pose a fire hazard. The lid looks OK, but the tank opening is not flat, so the lid won't seal. I filed it flat(ish) and fabricated a gasket using 1/16" rubber/cellulose gasket material. It still wouldn't seal, so try try again. I emptied the tank of fuel, stuffed a rag inside, and C-clamped it by the bar to a bench. This left the tank opening pointed downward, so material wouldn't fall in to the tank as I worked on it. I then took a palm-sander to the face with 600-grit emery cloth. That cleaned it up very well. Another close look at the plastic lid, and I could see my hand-cut gasket was actually a little too big to fit into the lid properly. It was curling up against the side of the lid in a couple of spots. Since I had to buy a 12x18" sheet of the stuff, I cut another one, this time being careful to make it slightly bigger than the threads, but smaller than the ID of the lid. That worked. The tank's sealed!!!
There was an o-ring for sealing the oil reservoir lid too. This also leaked, even with a new o-ring. I thought I was on a roll with the rubber/cellulose gasket material, so I cut one for the reservoir lid. This worked too. The big bucks I paid for the gasket sheet seem to have been worth it.
Somewhere in the process, I threaded the oiler trigger linkage through the saw, and attached it to the oiler body and the oiler trigger. After adding a little oil, it still leaked. It looked like the culprit was a screw in the tip of the oiler valve housing. The oiler piston pushes toward the screw. I removed the screw, wrapped it in 4-5 turns of teflon tape, and threaded it back in. Hey, it worked on the spark-plug. Problem solved.
Earlier on, I posted several pleas for help on a mystery hole in the bottom of the crank case. rmh3481 (see post #21) thought it might be something to do with impulse. I dismissed that, as it's nowhere near the carb. rmh3481 was on to something, but it's not impulse for the carburetor. I read the PM340 spec on Mike Acre's site. It says "Automatic oiler with Manual assist". There's no gear driven oil pump on this saw. I think, at some point, there must have been a diaphragm in there to use crank case impulse pressure to pump the oil. When I pulled the oiler body off, everything came off in chunks, and had to be ground off with a plastic wire wheel. What a mess. If there was a diaphragm in there, it was unrecognizable, and where would I get a new one now anyway? I guess the saw is now strictly a manual oiler. In truth, it was all manual before I took it apart, as there was no oil hitting the chain unless the oiler trigger was used. I'm less concerned about the plugged hole now.
So, it looks like it's ready for prime-time, warts and all.
When I find a little wood to put it through, I'll put up some video.