My controller works fine on the PS4 main screen. Can access everything displayed and all the buttons work fine. But, when I go into an app or game, I get no response from the controller. Have tried every suggestion I could find concerning the controller not working - reset controller, restarted console, unplugged console, deselected from Bluetooth and reselected, etc...
There could be any number of things that could be contributing to this issue.
As you mentioned, you have tried the default options, of resyncing, resetting, etc. Some other options might be:
1. (as provided by Wondercricket) Clean your Controller:
sounds like you might have a sticky or malfunctioning button or trigger, which is overriding the input given when you try to navigate through the game's menu.
I personally have had a similar issue that resulted in the controller exhibiting odd issues, and after sending it back to my local game store to get it cleaned (for a small fee) the issue no longer occurred.
2. Keep the Controller connected to the PS4, to see if that has any effect.
If this solved the issue, it's likely there is some kind of communication error occurring over the wireless connection. Again, this could be repaired by your local EB/GameStop.
3. Try a different Controller
Sadly, this problem might not actually be anything to do with the controller itself - it may be your console. Again, I have experienced an issue recently where the PS4 was acting very strangely - it was not communicating with the controller properly, and the game I was playing would not load. After some further investigation, the same issues were occurring with a different controller, and different games, and even to the point where it would "freeze" while attempting to trim videos captured using the "share" button.