My god, please stop this. You're spreading FUD. If Sony were to fix the automatic output setting (which they may have an nobody who knows wtf they're talking about has tested yet) this wouldn't be an issue, but to leave it at limited all the time even if your panel supports it is crazy. Blu-rays will have slightly washed out blacks because of Blu-ray not supporting 10-bit color profiles, but games will look worse if you leave ti as limited which, if you remember, lots of people use the PS4 to play games!
Not all sets have an option to turn on/off or even check the RGB support. Neither of my Sony TVs, one from 2008 and one from 2013, both have 10-bit panels and thus support full range RGB. Neither of them has an option for the RGB range, it just "does" full range.
Now you're spreading a bit of FUD. Blu-rays are not encoded in RGB and do not output in RGB natively. Your HDMI black level has ZERO affect on how Blu-rays are displayed. If a panel correctly accepts RGB full and limited no adjustments need to be made when switching between the two aside from adjusting the black level setting on your TV or monitor so it displays it properly.
To the TV comment that you own, I am not calling you out but that seems very odd to me that that model of TV accepts either full or limited and adjusts on the fly since it does not have a toggle. That kind of feature SHOULD be in all TVs so this incorrect display of black level nonsense would have been a non-issue but that is not the case. It your Blu-rays are looking washed out and you say you can accept full RGB properly on that set that doesn't have a toggle, that tells me that you used that slide as your reference, adjusted your brightness level so you can see the appropriate full RGB squares and now when veiwing content from Blu-rays which are encoded in the limited 15-235 Ycbcr range it looks washed out since your brightness is too high.
EDIT
What??? This isn't FUD at all, he's correct. Blu-ray movies are mastered in limited RBG, and if you set your ps4 to full and your TV to full theyll work great for games, but you'll wash out the range for blu-ray. Your games certainly will not "look worse" if you calibrate using limited RBG. They'll look perfectly fine as long as you calibrate with limited in mind for both ps4 and TV.
Again, NO. While Blu-rays' brightness/reference range is indeed limited i.e. 15-235 it is encoded in Ycbcr, NOT RGB. Plus, while its reference range is 15-235 the signal allows for whiter than white and blacker than black material to be passed and shown if the display and display device allow it. RGB does not allow this and if you force watching a Blu-ray to RGB this extra detail,granted it's not in all films/shows etc., will be lost as RGB does not allow this information.
Edit 2: If you properly adjust your brightness, i.e through instrument calibration or utilizing a test disc such as DVE Essentials, your brightness will be correct for limited and full range, RGB and Ycbcr. Your brightness should never had to be adjusted between the two ranges if your TV properly accepts both.