Yes, it looks like a good practical repair.

Sorry, I only got a quick look at the large photos before you deleted them, so I didn't really get to examine them in enough detail to give a sensible reply.
As Swake says, don't forget at least one cable tie to resist the cable being forcibly pulled out. It's really good that you've found that you can get adhesive lined heatshrink to fit over the connector at the far end. That should give you as much bend protection as a new grommet - probably more. Be sure to keep some available and replace it at the first sign of damage. 3:1 shrink ratio, adhesive lined, is common but I think I've seen 4:1 too.
Regarding the yellow wire. Judging length by eye, yes I think you are right that it was cut off flush with the original outer insulation of the cable, fairly common practice for unused cores. If everything is working correctly, I would just fold it back on itself and put a piece of sleeve or tape on it to avoid any possibility of shorts (to the metal joystick can for instance). That's assuming that it isn't also cut off at the plug end - better safe than sorry.
You've probably saved yourself quite a lot of money there. My neighbor managed to get a replacement joystick unit on ebay, but then had the problem of programming / configuring it to work with his chair. Irrc, it needed configuring for the options on the chair (lift, moving footrest etc) and things like acceleration. He was responsible for finding the bootleg configuration software etc. - my only involvement was implementing a non-standard USB-serial PC interface, but it certainly wasn't plug-and-play.
I'm glad you're back in business anyway. Try to be a bit more gentle with it - although I realize that it probably gets daily wear and tear. Maybe you can rig up something (even a bit of velcro) to avoid it getting dropped as much. Good luck with it.