Turning on PC PSU with current or bridging PS_ON and COM/GND?
Making a master PSU control a slave PSU is pretty easy.

Grab yourself a 2N3904 or similar transistor and a 22K resistor. Hook the emitter of the transistor to ground, the collector to PS_ON of the slave PSU, and the base via the resistor to the 5V line of the master PSU.

The only potential problem is whether the PSU and HDD start up fast enough to satisfy the PS4.