Except that what you proposed would cost more than replacing the frame.... A backyard TIG welder isn't exactly going to have a heat treatment furnace size capable where they'd want to bring this to T5 or T6, let alone does any of this mean you'd want to ride what's been mucked together and under any radar of security. Remove all components, clean paint, repaint, etc.

Rarely see it advised to repair alu frames unless they have significant sentimental value, definitely not for USD$700 off the shelf frames that have a lifetime warranty.