Troubleshooting power on problem...

I'm hoping someone can help diagnose what might be wrong with my PS4. When powering it on, it usually beeps, and shuts off within a second. Some specific details:

- it usually beeps a few times before attempting to power itself on (once for each time I press the button, where nothing happens until it's attempt to power on)

- once it attempts to power on and beeps and shuts off, it will not respond to further presses unless I unplug the power from outlet, wait a few seconds, and attempt this again

- it successfully boots up around once every 50 attempts. It used to power on more frequently, but it's getting worse.

- once powered on, it will be totally fine until I turn it off. It won't ever turn off by itself, even after 12+ hours on.

Based on my online searches, I concluded it might be the power supply. I ordered a (used) one off of eBay, and opened up the console and installed it. I still had the same problem , although the power on attempt beeps were happening quicker. It's possible this other PSU was also faulty, or it is a different problem altogether. Should I try purchasing a new PSU? Any other fixes to try?