I have a newer Toro Personal Pace mower (Model #: 20372, Serial #: 313018490), and I've had trouble with it starting and, when it does start, it surges. I figured out that I could set the throttle(?) with a twist tie, but eventually my mower would die a few minutes after starting it. I sent it to a small engine repair shop, and they put in fresh oil and gas and it started fine; unfortunately they didn't let it run for a few minutes so I still have the same problem.
The mower has been sitting all summer, and today I noticed that the part that I'm calling the throttle is no longer connected to the engine. I've attached a picture, and I'd be happy to take more if that would be helpful.
The part which you are calling the throttle, is actually the governor control arm. I don't see the governor rod behind your fingers, so I'm assuming it may have fallen back into the engine. Time to go back to the shop and have a talk with the owner, as you may have a major problem, meaning the engine will need to be opened up, for repairs to be made. The only way that part would come off, is someone forgot to tighten it up after trying to adjust the governor.