Hmm so you notice the screen is more responsive without?I pop a case on some of my phones - my Pixel 3 has the Google fabric case which I like. I've never yet found a screen protector that maintains the smoothness of the glass underneath so I tend to not bother with those personally.
I insure my phones from accidental damage now so don't bother with cases or screen protectors. Smashed too many screens not to bother.
Yes that's what I used do do, but from now on, I want to feel the screen naked, as it may be better performance wise. If you sell off your phones though, they get better value when they have the factory screen protector though.I keep them on if they're factory fitted.