Okay, I finally fixed it!
That absolutely cannot be the cause of that problem. Any damage to the controller will not affect the system in any way. Your PS4 might be having a problem outputting to your TV. Is it connected to it directly or through an HDMI hub? Cause I've had some hubs that had trouble keeping the signal on consistently.
Yeah, I think that was the issue.
First, I dissected the controller to determine if any pieces shifted around inside, then I pressed and held the tiny reset button in the back of the controller; still no dice.
Thankfully, I looked around on GameFAQs to see if anyone else had this problem and one person suggested to unplug the PS4, wait 60 seconds, then plug it back in. Sync the controller once the system is on, and it'll start working again.
That's exactly what I did, and it worked like a charm.