Okay, here are two reference pictures of the choke linkage and the throttle linkage. The choke linkage picture is just for reference, while the throttle picture is for the discussion that follows.
Throttle Drum and Governor Link w/Throttle Rod
The throttle link, seen above to the right of center, connected to the governor lever (GL) and behind the round "drum", is what I believe you're talking about. It moves up to close the throttle butterfly and down to open it up. Normally, at shutdown, even though the orange throttle cable may be in the slow idle position, the rod is pulled almost all the way down. To test it, run the throttle lever to SLOW and then try to lift up on the rod. It should have ~3/4" upward movement available. If you now advance the orange lever to fast idle, the throttle rod will pull down all the way, and you will find increased resistance to moving the rod upward.
Normally, when the engine catches, it will quickly move upward and keep the RPMs at ~1700. As you push the throttle forward to max idle, the drum rotates CCW (in this view) and puts more tension on the spring (behind the drum, not visible here) connected to the GL. This pulls the GL down, opening the throttle plate and increasing engine RPM. This continues until the engine RPM causes the internal governor mechanism to press down on the GL and balance the tension from the drum spring. That should be about 3350 RPM +/-.
When engine load increases (engage deck, push on hydro pedal, pull a loaded cart) the slowing engine will allow the GL to move down, opening the throttle plate, and bringing the RPMs back into balance at ~3350 RPM.
As I understand your problem, it won't start with the choke full on and the throttle rod in its normal, almost full down position. If you pull up on the rod, it will start. Does it keep running after that, or does it shut down after you release the rod.
If it shuts down, I suspect either one of the tension springs is missing or stretched. There should be two. One long spring connected to the idle tang and one short length, long crooked spring connected between GL and tang on drum. You can see both springs on page 114 in the manual referenced above. It is the lower right picture of the three on the page.
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