So you replaced the PSU? With a brand-new one, I assume? My assumption after doing extensive testing today (stripping everything out except PSU and back board) was that it must be the PSU, despite the fact that the back board's 5V Standby LED lights when the diagnostic button is pressed. Sometimes she boots, sometimes she don't wanna boot. I jumped the pads to power on and also used front button—it is not the power switch on the front board, as its the same when jumping it. When you start it she goes "click", and that's all you get. Then, interestingly, you have to hold the power button down (or jump it—same thing) for ten seconds to give it a hard power-down. Then, generally, after doing this cycle from 1-6 times or so, it will boot up. Is this the same for you? That indicates to me that a component either needs to be heated to a temperature where it will work, or it maybe is a capacitor in the PSU that isn't holding charge correctly and needs cycling to store it up (doesn't make a lot of sense), or it could be a defective relay. The caps in the PSU all passed at least a visual inspection—not saying much because a visual failure (ie, swelling or leaking) would almost certainly result in a total lack of power; although I have seen instances in automotive computers where a leaking cap just resulted in a weird behavior like one cylinder missing in the engine, which destroyed the engine... That cap wasn't in the power-supply side however, like most of the ones in the PSU.
Anyway, I am curious if you have this exact same issue, as it does strongly suggest the power supply. And, also, did you replace with a brand-new one or a used one? The only other parts to suspect would be the back board or the connection from the PSU to the back board... I did see some not-so-awesome-looking pads on the backboard, possibly indicating that they may have had some manufacturing defects.
Please let me know! Thank you. Otherwise I'll be tearing the PSU down again and going through it with a fine-tooth comb, which is a pain in the D when you just want a machine that works! Still, better than throwing money at a new PSU just to find that it doesn't work, either, and the problem lies on the back board somewhere.