Computer won't post / Beep unless Unplugged for a few hours
I'd suggest that you need to disconnect all internal peripherals (disk, floppy, USB (except keyboard) and test POST functionality for your motherboard.
You can get modern POST cards that will help you (here is an example: https://www.amazon.com/Desktop-Notebook-Motherboard-Analyzer-Diagnostic/dp/B01HRFQ8YM/)

If you test POST with only your keyboard and display attached you should either get Boot messages on your display, or POST failures on your tester card. 

As a quick test I'd ensure your +5 V standby is working correctly, I've seen this impact your ability to turn on a PC with no other side effects once started.