Due to how the metal has warped, I think this may be beyond repair.

Meh. Metal is a little like Play-Dough and nothing is beyond repair given enough labor time. But yea, unless it's unreasonably expensive or not available, even making a new one would be preferable than repairing that mess.

I've MIG welded and even stick welded stuff that thin plenty of times, but you do have to set the machine right and wiggle things around correctly at least. Looks like the guy just had it set too hot and/or lingered too long in the wrong spots. TIG is the proper way of course, mainly (in my opinion) because if you want you can set the machine so low it will barely light a cigarette so thin material is a no biggie.

Things are coming along. Looks like it will be a beauty when you're done.