650 Watts would be enough but yeah, if you're going for an RM750 you'll be sure to have plenty headroom and the unit should last you through a couple of builds/ upgrades. As a bonus the fan on it will probably never even need to spin.
If it just dies it definitely sounds like a power issue. Even if it's not the PSU it's probably the motherboard/CPU which you are both replacing anyway, so I'd be looking at the PSU for as the problem.
Also considering the hardware you're running on it, I'd swap out your current unit anyway, even without defects. 7 years is quite old for a PSU and I must admit I've personally never heard of the brand, but I couldn't find a lot about them on Google either which is always a bit dodgy to me.