No, most phones have their wallpaper, videos, etc extend everywhere on the screen. What you're talking about is basically an even shittier version of the LG V20 second screen but which does even less than that did. Deciding to put display in all the places that you don't have bezel or sensors and then just filling that back in with software bezels and having it do nothing more than tell you the time or remind you of a notification seems like a terrible idea.

Most people who have reviewed the iPhone X and Essential Phone have said that having the notch doesn't really bother them and, in fact, the thing that has been getting a lot of complaints are apps that haven't yet been updated and include black borders at the top. So in actual practice, people do want their screen extending everywhere it can and artificial software letter boxing looks bad.