Feasability of repairing '70s-era welded-steel luggage rack
That looks as though it was chrome plated after it was welded together, which means cleaning it up so you can safely weld it again is going to be tedious. You'll need to clean the chrome off anything that will get hot, and afterwards painting it will not look that great. It would probably be cheaper to build a new rack.

One repair option that occurs to me is to cut a T shaped piece of steel to fit on the bike side of the bracket, bending the bottom of the T out to support the lower part of the rack. Then make a second piece of steel for the other wide, probably rectangular, to actually hold everything in place. Drill holes in both parts to allow the mounting bolt to go through.

If you made that out of ~2mm thick steel, then painted it, you should get another year or so out of the rack. Assuming to can resist the urge to overload it.

A rack like that with one bolt really isn't designed to take much more than a journal and a baguette.