I've always used cases (Spigen) but I think screen protectors are a scam. The only shattered displays I've ever seen are broken screen protectors. I've only ever broken two screens and they were events violent enough that a screen protector wouldn't have helped. The other day I left my phone on the roof of the car. It came off around a bend at 60mph and apparently landed hard enough to remove the cover. Aside from a small chip on the corner on the back side it's fine.
I'm pretty rough with my phones; They're far more resilient than people give them credit for. When my iPhone 13 Pro Max gets dirty I wash it under a tap with hand soap.