The problem I've been told is there are no OEM parts available, so any rebuild is only as good as the cheap import parts. That $117 amazon starter looks like a good bet. Where's Vetech, still making the popcorn? A gear reduction starter will turn faster and cure many starting problems. You won't pay anymore for that one than a direct drive, but a genuine Denso, or Mahle/Iskra/Letrika will be $300 or so, and worth it if you plan to keep the machine forever.
You should be able to turn the engine over with the fan. Might have to grab the belt with one hand also. That will tighten the belt, and the engine will turn easily until you hit the compression. Let it sit five seconds and it will turn again easily. You should be able to turn the engine 360 that way, but since this has run fine, you just want to see if anything is dragging. You know it's not locked up.