Plenty of steerers have been extended by grafting, going back decades. Can be totally reliable. It is both easier and safer than replacing the whole steerer, even aside from the expense which is much less (and no repaint).
Back when steerers were threaded and quill stems had to go down inside, the graft would usually be put about halfway down. You just need the graft to be low enough to clear your "starfangled nut" or whatever you use to preload the headset.
It definitely can be done without any harm to the paint at the crown. TIG welding is faster, less heat input and less chance of paint damage, but it can definitely be done by brazing as well.
If you don't want someone learning on the job, give it to someone who's done lots of them. I know
Bernie Mikkelsen in Alameda Calif. has done a zillion of them, maybe give him a call?
I always encourage folks to tell us where they are in the world, especially when asking for frame repair advice -- maybe we can suggest someone who's closer to you.
Caution, about the stub you propose for reinforcing the joint: you say 22.2 mm, but be aware that a lot of threadless steerers are thinner wall for light weight, thus the plug needs to be larger than that. Measure to be sure (Apologies if you already measured).
No, the piece you add doesn't need to be Columbus. Just about any steel will work; it just needs to match the OD and wall thickness. Cut a piece of steerer from a dead fork. Huffy, Schwinn, it pretty much doesn't matter.
Mark B in Seattle