Reed valves that are not sealing properly can have an affect on starting, won't have any affect on the vacuum to draw fuel into the carb as they are supposed to open on vacuum producing upstroke of the piston anyway, but if they don't seal when the piston comes down it compromises the push of fuel up the transfers, fuel will pool in the crankcase and there will be spitting back from the carb. If you don't get spit back (wet filter?) when running, there is nothing wrong with the reeds. The slightest air leak anywhere in the fuel input path will reduce the ability of the choke to pull fuel into the carb. Can test this by removing the filter and pressurizing the fuel line, not likely a fault since the engine runs well, but easy to test. When the engine fails to fire under choke, you could remove the carb and check to see if the fuel chamber is full of fuel like it should be. If you change the control lever setting, be sure to pressure test the inlet as there is only a small range of adjustment until you introduce flooding problems.
Any restriction in the fuel supply path can also be the problem. Will the saw operate at full power with throttle full open and bar buried in wood? This is the situation that calls for max fuel delivery and if the saw will run this way for many cuts without becoming lean, there is no restriction.
