Hi AT3!
Yes, you could well set up the gear and drill and tap another hole in the boss. As close as possible to square to the bore, would be my advice.
You will have lost a little bit of bearing are when using Back Gear, but probably not enough to cause any disasters
You are in the "Chickem and Egg" dilemms. The really nice thing to do would turn away the boss where it bis broken and then to make up a Top hat bush, ready drilled and tapped for the grubscrew, and Loctite that into a slightly enlarged bore in the gear. But for that you need the facility of another lathe! Or someone nearby who can do those jobs for you.
As to removing a stuck chuck, there are are various slight variations on a theme.
Firstly, tighten the belts for motor and countershaft, setting up for maximum Chuck speed.
1 Grip a piece of hexagon bar in the chuck.Choose a sensible size, and for which you have a 1/2" drive socket.
(3/8 might do but 1/2 will ,be stronger )
2 Put a long power bar into the socket.
3 Hit the end of the power bar, HARD, with a mallet.
4 Alternatively, Apply an Impact wrench to socket, in the hope that it will hammer the chuck free.
or
A Grip a piece of sturdy bar (preferable to wood ) through the chuck jaws.
B Whack end HARD
Either way it might need more that one bash to free the chuck.
You may find that warming the body of the chuck with a heat gun will help it to unstick
In either case, hopefully, the inertia of the motor will be such that the chuck unsticks. Once it has moved a few degrees, the grip should decrease dramatically.
Place a board across the lathe bed, in case it fall!
For the future:
You already know not to engage Back Gear to lock the mandrel
Clean both internal and external threads.
DON'T run the chuck on under power, it will tighten under cutting loads.
HTH
Howard