The KIPS assembly in the cylinder should move silky smooth.
Try this-
Remove the claw gear (while supporting the shaft)
Confirm that the KIPS assembly moves nice and slick. A 10 mm nut driver through the left hand slotted cover would do the trick.
Start and rev the bike up a bit. The governor shaft should rotate through about 1/4 revolution.
This will help separate the issues-
Is the governor shaft rotating?
Now you can get an idea of how the claw gear should be positioned.
When putting the claw gear back on there should be marks that line up. If not, with the KIPS closed and the motor on, put the gear on so that it will open the valves.
is there some sort of spring in the cyclinder? or what makes it spring back?
There is a return spring down in the governer assembly in the right side case. It's not a very strong spring so your KIPS can't be hard to operate or it will stay stuck open.
Looking into the LH slotted cover , KIPS closed is fully CW. There ought to be a ball and spring that engages the cam on the shaft when fully closed. A lot of people remove the ball and spring; its purpose is to delay start of KIPS operation a bit. It comes up from below the lower edge of the slotted cover at about 5 o'clock.
From what you are describing it sounds like it is running with the KIPS stuck open.
You'll figure it out