I have never run into this problem before and I have worked on a fair amount of small gas engines.
The problem seems to be too much compression. The starter spins the engine fast with the spark plug out. With the spark plug in the starter spins till it hits compression and stalls out. The battery was replaced and I later returned to the parts store and had it load tested and they said it was good. I then focused on the starter.
I had 2 similar junk engines (Briggs and Stratton) with similar starters. tried both with the same result. Took apart the original starter to inspect the windings which looked good. Connections to the armature looked liked just crimped so I soldered the connections. That did not solve it.
When spinning the crankshaft by hand it takes considerable force to turn against compression. Compression gauge reads about 180 pounds sounds like a lot for a small engine.
The engine will turn over with car battery and car jumper cables put directly to the starter. Lock starter rotor amp meter shows 200 amps and battery voltage of 9-1/2 volts with new lawnmower battery. That is why I had the battery tested.
The John Deere dealer says this engine may have a compression regulator, which is something new and something else to go haywire. At this point I don't think a new starter would fix the issue if I could find one.