I worked in a phone/computer repair store, based on anecdotal research on glass screen protectors, this isn't necessary anymore.
I'm sure this applies to many phones (haven't tested specific ones) but the glass screen protectors refused to scratch for me, I tried screw driver tips, razors, as hard as I could against a flat desk.
If I applied enough concentrated force (or jabbed it) it would brake as expected, but anything short of that left a barely visible mark, if anything. I have seen them scratched, but it's not common.
I say throw a screen protector on it if you can and don't worry about it.
Having them in the same pocket I'd be worried about the force that could get applied by sitting down in a tight pair of pants with a modern-sized phone. The sort of situation that caused the "bendgate" problems. Maybe it's over cautious, but with phones costing $1000 I'd rather not mess around with it.
And I don't love the glass screen protectors, had one on my phone briefly and it seemed like it screwed up the image quality. Subtle, like a faint RGB shimmer over the whole thing. So now I've just got a thin plastic case and I try to not drop it.
What type of phone? You might want to try a different screen protector - every well applied glass screen protector I've seen on iPhones has been nearly indistinguishable from the glass screen (and I'm really particular about mine).
We did have issues on the Galaxy phones with that edge though, we could never get the screen protector to stick properly.
iPhone SE. I didn't get the cheapest garbage screen protector or anything either, it was a favorably reviewed one on the wirecutter. No bubbles under it. Maybe "slightly grainy" is the way to describe it? Been a couple of years so it's hard to say.
