It sounds like I have similar symptoms to this question, but this only started happening after swapping out a heatsink for a cooler. To be precise, the system will only boot if I press the clear CMOS button. This resets my UEFI settings, which is annoying since my RAID setting is reset (so my RAID is not recognized).
Outside of that, the motherboard LEDs are lit when power is connected. The power button just doesn't do anything (on the motherboard). Paradoxically, if I have booted the computer via "clear CMOS", I can turn it off with the main power button. Any thoughts?