Similar problem today with my GC2300 which also has about 700 hours though not sure what year. Exactly the same symptoms as the original post here, though my battery was only 12.7v when checked with the meter.
Tried to jumper from the positive battery post to the positive terminal on the solenoid, but no result. Tapped on the solenoid a couple times with my pliers and it started immediately.
The next time I stopped to get off, I left the tractor running, though in neutral and with the PTO disengaged. As soon as I stood up, the tractor shutdown. After some fiddling I've determined that the PTO lever or the switch for the PTO is off kilter somewhat. A few times I had to hold tension on the PTO lever in the disengage position to get it to start. A couple times I did not. The lever doesn't seem to be coming back as far as it normally does, so when time allows I'll blow out the lever linkage and switch with the air compressor. It's been rather dry & dusty the last two months here.
I might not also have a starter solenoid going bad.
Thanks to all for the advice! It was a big help and I was only out of service for about 30-40 minutes.