@Cyan had the exact same thing happen after a power surge. I think he ended up replacing it.
Definitely interested to see if you can manage to fix it.
a little late for a reply, but yes I had the video issue where I still could use the console normally (I could hear the menu, and game launching, etc., not only the recovery).
As I don't have enough knowledge or technical power into replacing any chipset to fix the hdmi output, and seeing there's no hdmi module but everything directly on the mother board, I ended buying a PS4 Pro.
The process to transfer the data from one console to the another was surprisingly very easy and didn't even need to plug the old console to TV, just press power button !
I could recover all my data on the PS4 pro. I still have that old PS4 I can't use and don't know what to do with it. maybe one day someone will know how to find the fried electronic element and fix it.