Hey OP I've been dealing with this problem lately on my 3 week old PS4 and I think I can help. Check my comment history and you'll see I talked about this problem this week and someone gave me a tip that worked and I was really excited but it only worked temporarily though.

However I finally found a fix for my issue yesterday and my problem is almost exactly the same as yours and I may be able to help. I had several different blue light problems all of them power failure related, never once overhearing problems. I will type more later when I'm not on mobile. Sometimes the blue light would only stay on for 2 sec with no fan and die, other times the fan would even start and the hdmi input would almost change but then it would die after like 6 sec. If you tried to start in safe mode and actually got to the second beep it would shut off before starting.

Try unplugging the power cord from the wall and PS4 and plug everything back in. Hold console's power button down for safe mode with the controller already connected by USB for safe mode, and as soon as you get the second beep at 7 seconds for safe mode immediately release the button. This allowed the light to turn white and turn on the console and do the "checking hard drive status" percentage bar. Sometimes your console will die before the progress completes: it will take a few tries but follow the above steps to get it to turn on again and it will usually complete.

Once in safe mode try to rebuild your database and hopefully this process will make it work. I know rebuilding the database is sort of a hard drive issue but somehow this got tnworking for my power issues. Hope this helps and happy to explain more later.