Hi!

Just out of curiosity, have you tried powering the system without the
battery installed?

I don't think it's a problem with the battery or charging circuit. It
can't hurt to see, though. It really sounds like something on the
motherboard is shorted, overloading the DC-DC converter inside the
computer and causing the power brick to shut down.

If you want to fix it, I would start hitting eBay for replacement
motherboards. It may be cheaper to get a machine with a busted display
or something than to get a motherboard.

William