The power supply provides permanent +5V on its small connector. This +5V is used to power the components responsible for supervizing the power button and giving back the info to the PSU (Power Supply Unit).
When you touch the power button, the PSU is activated and provides +12V on the giant 2 leg connector.
In your case, it seems that the +5V is not produced by the PSU - or - the components powered by the +5V are malfunctionning.
I would first dismantle the PS4 (assuming the Eject button is not responding either), check the +5V with a multimeter, then check for any fuse on the motherboard.
If you find a blown-up fuse, please, please, do not bridge it or you'll fry the motherboard, just report it here and we'll see from there.