I don't think it is worth buying a tachometer just to set the maximum engine speed, which is probably close to correct anyway if the governor is undamaged. On the GCV engines there is no adjustment for maximum speed except choosing between the two holes where the governor spring attaches to the bottom end of the governor arm. Your engine seems to be set for the "normal" maximum speed, the other hole in the arm would give a higher maximum speed. I suggest you leave the setting as it is.
By way of background, the GCV series of engines appears to have been designed as a reduced-cost series, after the GXV engines had been in production for a time. The GXV engines have pushrod valve gear, detachable cylinder heads, and lots of adjustments and oil seals. The GCVs don't. The GCVs appear to be what I would describe as throwaway engines: they have a rubber belt driving the camshaft, and when it stretches and slips, the crankcase has to be dismantled to replace it. Valve lapping and piston ring replacement are difficult compared with the GXV engines. None of this is very relevant to your problem, however: a GCV engine should last quite a few years in normal suburban service.
There is a possibility that your engine is fitted with the optional spark arrester, and if this has become clogged (for example through the use of dirty or inferior fuel) it is possible the excessive back pressure would cause symptoms of lean mixture, such as you have. I suggest you follow the instructions on Page 83 of the manual, and if the spark arrester is fitted, ensure that it is clear.
Look at the pilot screw (Page 77). Essentially this is just an idle mixture adjustment with a rather limited effect, and on current engines it is normally not adjustable anyway: a movement limiter is attached to the head of the screw. If it has the limiter, you are probably stuck with that, because it is not appropriate for me to tell you, and everyone else who visits this site, how to defeat it. Note: do not try to remove it by force, it is deliberately designed to break the brass screw off if you do that.
Please post a picture of the float bowl area of your carburetor. I will then go through the process for cleaning the main jet and emulsifier.
