Can Running a Server's Redundant PSU on Different Circuits Damage It?
With multi-phase AC systems (motors, etc.), you're right, bad things can and will happen if one of the phases drops out. However, with computer PSUs, each of them operates completely separately, converting its AC input voltage to a variety of DC voltages for the computer system.
You can safely run redundant PSUs on different circuits, different phases, etc. Doing so is actually a really great idea to reduce the number of components that are fate-shared.
The job of the power supply is to take power that is not 100% clean and make is stable.
People working in data centers have been fired for not having the power supplies run on different circuits. If one circuit breaker goes off, or one UPS dies, the whole server goes down which is not the intended design.
Ideally you would want them on separate circuits on with a different UPS on each circuit so that if anything goes down the server is still good to go.