How to troubleshoot a black screen (no video) on boot

I have fixed problems like this in the past by replacing the power supply. I had a computer that would randomly completely shut off with no warning. Linux, in this case, but as you said, the OS probably doesn't matter. Replacing the power supply fixed it and the problem never recurred.

Although power supplies have no moving parts (except for the fan), they wear out surprisingly often. I'm not certain how one could prove that the power supply is the root cause of the problem other than swapping it out and trying.

I try to always have one spare power supply on hand so I can at least test this without investing in a new power supply. The spare is usually one of my working but older power supplies. That is, my spare is not a new, unused power supply, since this is just for home use.

I have a multimeter, but I've never used one before. How can I check to confirm or rule out the on/off switch? Are there any other items I might want to check?

Turn your multimeter to the Ω/Ohm/Continuity tester. The Ω represents the amount of resistance in a circuit. A piece of wire (closed circuit) should have a resistance that is near 0 ohms. Air (open circuit) will have a very high resistance, most likely infinite from the perspective of your meter.

A power button on an ATX motherboard is a momentary switch. That means the circuit will only be closed when you press it.

Here is how I would test. Find two paper-clips and insert them into the motherboard connector for the power switch. Make sure the paper-clips to not touch each other. Touch one of the probes to one of paper-clips and the other probe to the other. While you have both probes contacting the paper-clips press the power button. If the button is good the reading on your multimeter should go from infinite ohms down to about zero.

BTW, if your case also has a reset button, you may be able to use that as a replacement for the power switch.. On an ATX case the reset button is just another momentary switch.