I think the problem with today's screen damage is different than the day of palm pilots. Back then a sharper fingernail could scratch the screen; now you could use the screen to file your nails. Now the damage is more about cracking. I don't think a plastic one will help in any way against it, but I have seen phones with tempered glass screen protector where the glass protector cracked but not the screen. It is probably because the protector cracks easier, and even without the protector, the phone screen would not have cracked. But it does make you feel better for having the protector on!
Having a case is more important though. My nieces and nephews who do not like to have cases have cracked screens... all of them. They put their phones in the back pockets of their skinny jeans and sit on them.. it is like a torture test for phones.