I would first check the throttle cable, does it move the throttle plate lever which is attached to the governor spring and pulls on the carb butterfly fully open?. The RPM is governed by flywheel air pushing on pivot blade which pulls on the spring, this pulls on the carb. butterfly to choke off some air so the engine will not over rev.

Sometime the metal wire inside the coiled throttle cable will slip at the hold down clamp (or be binding or broken). when you push or pull the throttle lever it just slips at the carb and don't put tension on the govenor spring which won't open the carb. butterfly, which won't cause the engine to rev up.

If you can pull on the governor spring and the engine revs I would suspect the throttle cable not advancing the throttle plate lever and not pulling on the governor spring. All of this is providing you have the carb. connected in the right holes to advance the carb. butterfly valve to let more air in and the throttle cable is advancing the butterfly valve open.

The more common thing is that the front tab for the governor spring gets bent in so the tension on the spring is low.