I had the stator cover off and the gear shifting box to make sure I hadn't damaged anything when I broke the connecting rod between the foot shifter and the gear selector.

When I put it all back together, it wouldn't start! The solenoid, starter, battery, gas, air, etc are all fine. Come to find out, I had put the stator cover on too tight, so I took it off and tried it again. Everything spun correctly. I then put it all back together (being careful not to over-tighten the cover bolts).

Now when I start it, it sounds like it's sticking a bit. It will rotate a little, then stick in a spot and then rotate some more, but it won't start.

Any ideas how I can fix this?

You might have the washers installed wrong.
Both the washers go on the spur gear with the torque limiter.
BTW: Can't listen to your vid at work, making an assumption you have a post 95 bike.