Thanks for that very clear report, Jaffa.
First, it seems certain that your lean mixture problem stems from a fuel cap malfunction. I suspect that by screwing the cap right down without that spacer piece that had fallen out, the fuel cap vent did not operate. You may find that you can re-attach the loose bit and get it working again. Of course a new cap would work even better.
Second, those in-the-crankcase centrifugal governors mostly work in much the same way. It is important to set the angular relationship between the governor paddle in the crankcase (which bears on the plunger that is pushed out by the centrifugal weights) and the governor arm on the outside of the engine, because the plunger has a very limited operating range. So, you rotate the paddle's shaft to bring the paddle into contact with the plunger with the engine stopped and the plunger fully retracted. Then you pull the top of the governor arm to the full throttle position, and lock the arm to the paddle's shaft. This means that no plunger movement is wasted. The plunger's role is to push the throttle toward closed, with an amount of force proportional to the engine's speed. The governor spring's role is to pull the throttle toward open, with an amount of force proportional to the speed control setting. So, if you set the speed control to maximum, with the engine stopped, the governor spring will be fully stretched and will pull the throttle firmly into the wide open position. When you start the engine, the centrifugal mechanism will push on the plunger, trying to rotate the paddle's shaft and move the governor arm toward the closed throttle position. The governor spring will keep trying to open the throttle and the paddle to close it, and they will fight each other to a standstill at some engine speed that the manufacturer decided should be the maximum. The speed is usually about 3,000 rpm for mowers, so the blades won't throw stones dangerously hard.
I suggest that you fix or replace the fuel cap, set the governor linkage as directed in the workshop manual, and then give it another run.
