Power Supply for 2.5” HDD – Requirements and Options

It's theoretically possible. Switching power supplies (like a PC PSU) all have a minimum load requirement, typically on the +5V rail. Perhaps there was a device you forgot to plug in on the +5V/+12V rail?

It is perfectly possible to measure good voltages and still have a bad powersupply.
You suspected that already...

The thing is that a multi-meter is a relatively slow device that won't pick up on very short power-spikes or dips, but these can easily cause a reset of the harddrives.
(There are some multi-meters that can do this, but they are very high-end expensive ones.)

The motherboard is generally not affected as it has it's own voltage regulators that provide some buffering and filtering, thereby isolating the circuits on the motherboard from the worst.

Without a second power-supply there is really no way to tell, but a cheap power-supply, just for testing, won't cost much. $25-$40 will probably get you one. If the system is stable with that you know for sure it was the power-supply. If it is still unstable you have a spare for future use.

But something else: Are you sure you are not overloading the power-supply or running it very close to it's maximum capacity ? 350W could be a little low for a server that also does VM's.
Especially with USA low input voltages (122 V) some power-supplies have serious issues if any of it's output rails needs to deliver more than 85-90% of it maximum rating. Check the label on the power-supply for the max ratings per voltage and do a little calculation what you are powering from it. Just to be sure.