Good points. If the original carb is still available (a WT22 maybe?) I'd be going back to that for starters.
And to break it back down to basics, there's either too much or not enough fuel at idle. The tipping forward clue could indeed indicate that the saw is loading up at idle and simply flooding itself out. Should be a pain to restart without pulling it over a LOT. Pull the plug and see if it's soaked with fuel. Turn the saw upside down and pull it over to see if fuel comes out the plug hole. If we have an overly rich condition due to the L screw adjustment, crank the idle adjust screw in for a stable, but high idle. Then adjust the L screw in each direction until the rpm's drop at each extreme. Find the middle ground between the two (usually the highest rpm) and call it close enough for the moment. Then adjust the idle screw back down for a normal idle (or at least until the chain stops). Rev and release a few times to see if it will settle into a reasonably normal idle without stalling. If it still stalls, it's still either not getting enough or too much fuel at idle. If the L adjust screw is somewhere around 1 to 1/2 turns out from seated, we're gonna rule out the L adjustment as the problem for the moment. We're also gonna check the plug again to see if it's soaked or dry. If it's soaked again, we're now going to investigate the metering situation of the carb itself. Odds are the metering lever is set way too high or there's debris at the seat of the inlet needle, either of which will allow too much fuel through the carb despite the general L adjustment we just roughed in.
If on the other hand if we have a dry (or normal) plug and the saw maybe even needs to be choked to restart after it dies, and given the fact that it seems to run and cut fine at normal operating rpm's, we're not going to go through all of the usual possibilities for fuel starvation like air leaks, fuel line issues, etc. We're going to look at the metering situation again after carefully inspecting the orientation of the diaphragms and gaskets to make sure they're assembled in the correct order..., just because. If the metering lever is too low, there won't be enough fuel drawn through the carb at idle even with the L screw turned out as far as you seem to have it set.
Depending on the knock-off carb you're using, the correct metering lever height could vary from the 'general' norm which is either level with the carb body or level with the floor of the metering chamber. There are exceptions in both cases, but the 'norm' for either carb body type should at least be close enough to allow the saw to run and idle. Both Walbro and Zama have a tool for setting the correct metering lever height if it needs to be below the 'norm' .
https://www.google.com/search?q=2+c...ome..69i57.10779j0j7&sourceid=chrome&ie=UTF-8
The easiest test for all of the above would be to simply change the carb and adjust it accordingly (provided you know it's clean, the diaphragms are cool, etc). If it is indeed a WT-22 like I seem to recall, it is a more than capable carb and certainly worth the time and cost of rebuilding. Unless it's filled with rust and corrosion it should come back around with a kit and some carb cleaner.
And speaking of diaphragms, it's also possible that the pump diaphragm in the current carb is too stiff to draw fuel at idle with the available impulse strength. There
are different types of pump diaphragms made from different materials. The amber Teflon diaphragm is generally the thinnest and most flexible.
So....
Go fix this freekin' saw, dammit!!! I'm gettin' tired o' typin'!!!
LOL