Damaged PS4 output?

Hey, I usually hate posting about personal troubleshooting problems, but this has me stumped, I haven't seen this anywhere else, and I trust the advice from this subreddit.

TL;DR in advance: PS4 seems to operate fine, but the output isn't working. Tried different TVs, HDMI cables, and starting in safe mode with no success.

It seems to me that the HDMI out on my PS4 stopped working suddenly. Earlier today I used the PS4 with my projector as usual. No issues. Later, I usedmy PSTV to connect to my PS4 for the first time for Remote Play. It worked fine, but I couldn't figure out how to end Remote Play, so I simply closed the app. Shortly after, I tried to hop on my PS4 and it seemed to already be on, probably because I never shut it off in Remote Play. But this time, it wouldn't display on my projector or play audio through my speakers. I was getting No Signal. I powered it off blindly and turned it back on. Still no signal. I swapped the HDMI cable. Nothing. Tried bypassing my receiver and going straight to the projector. Nothing. Finally picked up my TV upstairs and carried it down to the living room to test it there. Still nothing. I started looking up solutions. The obvious steps aside (loose cable, crap in the HDMI port, bad cable, etc.), all I could find is to try booting into safe mode. From safe mode, apparently you can change the resolution and try to fix it from there. I gave this a shot. The No Signal message disappears, then it seems to find something, but it's just brief static before going back to nothing, then static. This repeats a few times before it returns to a blank screen.

Aside from not seeing anything, the PS4 does seem to boot properly. I can access it from Remote Play.

Has anyone experienced this or know any troubleshooting techniques? I really thought this was related to Remote Play, but it seems to just be display connection issues. Either that or my PS4 still thinks it's being accessed through Remote Play somewhere and it's disabling the output.

...If all else fails, does anyone know how I would go about transferring save data from this PS4 to a new one? I'm 105 hours into Persona 5...