Unfortunately most home use PSU's have no self test build in, so you need to verify it yourself. Are the voltages in range? also under the heavy load? Almost all other units in our PC do have a POST (power on self test). So if a PC boots fine, what remains are mainly intermittent failures, which are a lot harder to uncover.
Apart from hard drive failures, the three most common types of failures you'll find on a mother board are:
1) faulty supply capacitors. These cause supply noise, or even worse, make the supply on the board itself unstable. The small ceramic caps on a board rarely fail, but the bigger elcos often do. The result can be that digital communication on the communication busses fails, or parts become unstable.Not so easy to diagnose by software. What is a clear sign, when more then one part on your board seems to fail. With a scope you can check for supply noise. Some bad elco series were easy to see, their packages were ballooning and they were leaking some brown stuff.
2) intermittent connections. This happens more often then you would guess, but certainly in modern PC's where big (memory) chips are soldered down on ball grid arrays, failing connection do occur. To diagnose it, you could try tapping on the board with the handle of a screwdriver, while your PC is running some heavy program. But that does not reveal all bad connection of the ball grid arrays. More effective is "cold spray", cooling the suspected parts down will widen the gaps of bad connections.
3) Faulty chips. I think this is actually rare, chips die pretty hard (even memory), most parts suffer from failing connections. But if a chip is on the edge of failing, that is revealed by heating them up. Give them a stress test using a heavy load, place your PC in a warm place, direct sunlight and run a stress test. This is probably best captured by software tools.
"Consequences, Schmonsequences, as long as I'm rich." -- "Ali Baba Bunny" [1957, Chuck Jones]