I as well recently been having a very similar issue…
On initial start up of the PS4 I would get a system error indicating “Cannot Start PS4. Connect a controller and press the PS button”. I messed around in Safe Mode multiple times trying the different settings and even after a couple times choosing option #7 “System Initialization” the PS4 would then work properly for about a week before returning to this cycle of errors.
Finally I pulled the hard drive out and plugged it into my desktop PC and it did report some bad sectors, I wiped it and reformatted it (exFAT) and it reported “good” condition but still overall no luck.
Lastly I used a different hard drive entirely and this is where I am stuck. Upon booting into Safe Mode the only option I get is #7 which is a full system initialization. I have verified the correct file I put on the USB key which is the full system file and not just the update file (version 7.51). The PS4 reads the file, asks if I wish to continue because everything will be deleted and I proceed.
1) Step one, it does get to 100% full system initialization.
2) Then proceeds to do a system file update which succeeds (safe mode window)
3) then it displays PS4 graphics screen and does a second system file update and succeeds
4) then it reboots and does a system storage install which actually takes a bit of time to complete.
5) It’s then after it completes the system storage like building the database where it does a final reboot and then it displayed the PS logo but then just sits on a black/dark blue screen. The “blue” system light is continually pulsing if but will just sit there as long as you let it. I have pressed all the buttons and so on only thing left it to turn it off.
Upon turning it back on I get the “checking system storage”, it gets stuck at 24% and then other errors again about not able to start. If I go back into Safe Mode at this point I am back at square one with only option #7 as my choice as if it didn’t actually load anything we just saw as successful.
I will try a brand new store bought drive to eliminate the possibility of a bad drive but after this without dismantling the entire PS4 I have no idea how else to fix this PS4.