Remove the starter and hook up the electrical connections to it. Ground it to the engine or tractor frame with a battery jumper cable. Turn the key to start. Does the pinion gear move outward (or at all)?
We tried that and the gear spun like crazy but didn't make an attempt to go up. When we got this mower we had to put a new pinion gear on it and it worked great, but when we would mow it would mow for a little while and then the blades would shut off. Someone told us this was due to a weak battery. So we replaced the battery and when the new battery was installed the starter would spin but the gear wouldn't come up. So we replaced the pinion gear again and the starter drive and still does same thing. ( It also has a brand new solenoid, clutch switch and ignition on it.) I really don't want this deere to become a flower planter but my wallet can't take many more replacement parts