I'm 100% certain that your old power supply is fine. If it would be broken or not sufficient to run your GPU and so on, your PC would make a starting noise but never actually start up.

Rebooting is usually the cause of overheating of the CPU or (a broken) motherboard. I don't know what kind of RAM you have installed, but it might be a good idea to check those out too if you're able to.

When it's not rebooting at the moment, does it load your harddisk without problems?

Cheers