USB supplies power and significant amps, enough to kill a low end or dying PSU but I understand the external power source.
Without having the machine in front of me, that seems the most logical. I wouldnt plug that PSU into it in case it is passing out bad power and could fry your new board. Honestly, might be cheaper and easier to take it to a local PC shop just to see what the issue is. I know the one I worked at would diagnose for $20. At least then you have a second opinion. Typically the only things that keep a PC from powering on for at least a second are, PSU, CPU and motherboard. Even without RAM a motherboard will at least power on.