My experience is a carb cleaning is usually necessary after running out of fuel. A quick test wouldn't hurt. Unplug the PTO at the connector closest to the PTO and then try and turn on the PTO and see if the engine still stalls. If it stalls you know that it's a issue in the safety system, if...
This style stator brake was most likely rushed out to meet new safety standards, stopping the blades faster when the safety switches cut the engine. The OP is now finding out why newer version of this have their own winding on the stator for the brake. I'm betting the relay failed while he was...
Here is a schematic that includes a startor brake. The model number you listed doesn't seem to include one in it's schematic, maybe there was an engine or wiring harness swap in its past.
It's probably a stater brake designed stop the engine quicker when powered off. They were fairly common on these kohler equipped craftsmans of this era.
You seem to have posted the wrong model number, or sears is giving use the wrong one for that number because it doesn't match the pic at all.
There is no adjustment for the clutch on those. If it is still running the trans with the clutch pushed the only thing I can think would be a belt that...
I would check the valve lash before condemning the starter. Sounds like it's out of spec causing the compression release not to work. It rotates the engine until it hits the compression stroke then gets it started on the next rotation where it gets a full running start on it.
Time to check the right drag link and the tab where it attached to the axle. It's pretty standard for these to get bent and throw off the alignment on these machines.