A screen protector is just to protect from scratches. Not from drops. If you drop it chances are the screen might still crack from the impact.
Thinking about putting a screen protector on my car and hope if I get into an accident nothing breaks
You actually do have a screen protector on your cars windshield. All US cars have safety glass installed.
As for screen protectors protecting your screen, some will do more than others. Specifically, wet application screen protectors can potentially protect your screen from what would otherwise be a broken and cracked to crap screen, to something very minor. Wet application increases the screen ability to absorb more flex before breaking. Think of window tint on your car or special hurricane 3m film used on windows in the south. The glass can flex much more than without the film. And of course, when it does break, all the pieces remain in place (majority of the time anyway. This could be the difference of a working or non working phone after a drop.)
Obviously, this won't completely protect the phone from all impacts and not as good as a plastic case, but it is an additional layer of protection.
Dry application screen protectors are worthless other than from key and coin type minor scratches.
Biggest thing, treat your phone like a baby! Don't drop it.