Not much you can do

Not much you can do, since it definitely sounds like a hardware issue. You can also add to your list that the power button is sticking intermittently and that the cable transferring power between the MLB and PSU is bad. So, you basically just have to start replacing parts until you figure out what's wrong.

Given that it works at least sometimes, I'd be putting my money on the MLB being bad. Though you could take it apart, disconnect the cable for the power button, then just short the pins to turn it on. If the symptoms go away completely, then you're left with a bit of a conundrum. The rear housing is a b itch and a half to replace, just because of the sheer amount of effort involved, and especially on those old white models with that annoying as crap EMI shield attached to the display... But it's also relatively inexpensive.

What do the diagnostic LEDs show when you plug it in but it won't boot? One LED means that the PSU is supplying power, two LEDs means that the system is booting, then depending on the model there may or may not be a third, which just means that it's communicating with the video card, and the final one is that the video card has initialized the display. It's also useful to have the thing taken apart when you try pressing the power button to see if there's any signs of activity when it refuses to turn on. Sometimes the fans will run but it stalls pre-POST, and that's an obvious MLB issue. Take the bezel off and get the EMI shield out of the way, and run it like that until you can duplicate the issue.