Remember how a screen protector works.
There are tiny (microscopic and smaller) particles floating around in th air. If one of them lands on your screen, and you wipe the screen, it's going to scratch even Gorilla Glass 3 (which has a hardness of only 6.7 or so - carbide, and there's a lot of it in the air, has a hardness of 9).
Dropping the phone, screen-side down, on a stone, will shatter the glass, whether it's bare, has a plastic protector or a tempered glass protector.
Dropping the phone at just the right (or wrong, depending on how you look at things) angle will chatter or spider the screen glass. The harder the glass, the easier it is to break from shock. (Diamonds can be cut [cleaved, actually] with a soft iron blade, if they're hit on a cleavage plane.) If you drop the phone so the shock goes through a cleavage plane of the glass, it will cleave, no matter what kind of screen protector it has on it. So you want a case that will absorb the shock and a protector of some kind that can be replaced when it's too scratched for your comfort.
I don't work for Otterbox, nor am I connected with them in any way, but for non-waterproof cases, it's the best case I've found. It absobs shock and has a built-in plastic shield over the screen. It's more expensive than a plastic or tempered glass shield, but it does a better job. (Go to a store that carries them, take one out of the box and hold it in your hand and you'll see the kind of security it feels like.) I tripped over the charging wire on my phone this afternoon and the Otterbox case put a dent in the coving at the bottom of the wall on the other side of the room. The Otterbocx and the phone are fine. (At least the charging port seems to be fine - that's about the only part that could have gotten damaged.)
If you want waterproof too, get a box of zip lock plastic bags for a couple of bucks. Two bags, one inside the other (zipper-side of the first one in), makes the phone pretty waterproof, not just water resistant, and it's a lot cheaper than an $80 case. (And you can use the phone through the bags.) It may look funny, but the first time you drop the phone into a lake, it's everyone else who'll look funny - when you're unconcerned and use your phone as if nothing happened.