Motherboard – Understanding the 12 Volt Power Connector

From that specification sheet, it should be compatible with any modern computer on the market.

It has many outputs and then some... Seems fine by me.

What you described to me looks like some sort of safety system in the power supply.

I recommend that you disassemble whole computer and check for any burn marks on the motherboard and pretty much everything else. There could be something which causes a short and makes power supply turn off. Pay close attention to computer case and check if it's moving or vibrating.

Next step would be to check cables in your power supply. If you have multi-rail power supply, you could be hitting rail limit for power. Make sure that load is distributed among cables and devices. If a cable has several connectors on it, try to use up all other cables first before using extra connectors. Make sure that you balance load on cables with several connectors. For example on a cable going into a hard drive make sure you only connect low power consumers and so on. This in general shouldn't be a problem, but some power supplies can be picky and may have bad load distribution.

Next check how different devices are mounted. I once had a similar problem. I bought a new computer and every once in a while it would shut down unexpectedly. In the end it turned out that the incompetent or malicious manufacturer used too long screw on the floppy drive. They penetrated inside and caused shorts from time to time. Make sure that nothing similar is happening in your case.