Did you use a Multimeter to make the determination that the ATX plug was good, and the other power connectors were bad?
If so, what were your voltage readings? (for example: +5.00vdc, +12.00vdc, -5.00vdc, -12.00vdc, etc.). When PSUs fail, and internal components fail, it can overvoltage the components getting power from the Motherboard--for example, case fans and lights. They can do unpredictable things with both overvoltage and undervoltage to them.
I had one couple of months back that was not reading correct voltages, so I ordered a replacement (HP), and plugged it in and it appeared to have correct voltages on Multimeter. Until I connected Mini-ATX connector to Motherboard, and lights blinked on and off instead of staying on steady (power LED), and the case fans spun up, spun down, and then turned off!
That's a sure sign the Motherboard circuitry has shorted out. The place I got the PSU from is an authorized HP parts reseller and they test 100% of their PSUs on load and no load.
Sounds like the Motherboard is at fault, since you threw a 2nd known good PSU on it and it failed to operate. Sometimes, you'll get lights, no lights, fans, no fans, or a combination of all as per my story above.
ThinkCentres are IBM and made really well. You might consider Motherboard circuit repair; they run around $100; I personally don't use those services but I know Techs who have with limited success. If you can't afford to lose the $100 I wouldn't do it, it would either junk it and use for spare parts, or try and find a Refurbished Mobo on eBay and re-purpose it.
BIGBEARJEDI