That sucks. Yeah, similarly avoiding Gamefly after they sent me a cracked disc. You mean the game won't even start up? The PS4 doesn't let you install games entirely (I wish it did like the 360).
It does start up, actually -- it just stops at a certain point during the background installation, meaning that the game becomes unplayable a certain point after the tutorial mission. Watch Dogs 2 has a "Global Installation Progress" meter in the Main Menu which tells you when the game is fully installed. In my case, it just stops at a certain percentage and won't continue regardless of how long I leave the game running (and there is no noise from the disc drive, so it is not even attempting to copy anything). I tried various troubleshooting steps (rebuilding database, installing with and without the update patch, installing on another PS4) and it stops at the same point, so I can only assume that it's a problem with the disc.
The PS4 does actually install games completely, too, though you usually have to leave the disc in there a while before it copies everything and it won't tell you when it's done other than the disc not spinning anymore (unless it has a meter like Watch Dogs 2 does or requires you to wait for every file to be copied before allowing you to play). My process for completely installing most games involves putting the disc into the drive, letting it complete the small installation which is required to start the game, then sitting at the main menu for anywhere from 15 to 45 minutes. At some point, the drive will stop spinning, indicating that the system has copied everything it needed to from the disc and from that point, it will only spin the disc when it is first inserted to read the license and start the game then use the files copied on the drive to play the game.
That's my understanding of how it works, at least, and I usually go through that process instead of just playing the game while letting it install in the background because I'd rather give the system time to finish with the disc so I can play uninterrupted without waiting for something to install. It also, in the case of GameFly games or used games in general, tells me if the disc is good because if it isn't, I usually get a blue screen error at some point during the background installation telling me the disc is damaged or dirty when it reaches a part of the disc that can't be read. I am not sure why Watch Dogs 2 isn't giving me the same error message, but from what I have read, the game should normally continue to install until it's finished.
It could have just been a bad disc from the manufacturer, as I have read about others with the same issue who bought the game new, so I cannot say with complete certainty that it is due to GameFly's (or their rental customers') handling of the disc but the fact that every disc I've had an issue with since purchasing a PS4 about 2 years ago has been from GameFly hasn't made me very confident that they aren't to blame for some of the bad disc problems or at the least, don't do very thorough quality checks when they get the discs back.
