Well that's good news, at least!
I must say you're not really going about the repair in the right way. Usually you don't reverse-engineer the entire board, you troubleshoot individual sections to find where the problem lies. That, of course, comes after a thorough visual inspection of all parts.
How about starting fresh: What is the welder doing when you try to turn it on? Is it completely dead? If so, chances are it's a problem with the power supply. That's usually the best place to start.