Is there a launcher or some software which would ignore top 1 cm of the phone screen?

When I had this problem, I think I learned from this answer.

One option is to use adb, i.e., the command line client/tool that comes with the Android SDK. It does a variety of things, but the thing that is most relevant here is the ability to use it to reconfigure your display.

Specifically, you want to access the shell commands to change the window manager's overscan setting with adb shell wm overscan L,D,R,U.

So you will want to use adb shell wm overscan 0,0,X,0 to push the top of the screen down by Xdp.

You can revert to normal with adb shell overscan wm reset or adb shell overscan wm 0,0,0,0.

If you don't know how to adb, it doesn't take much to get to the point where you can type that command in. I can relate because I used to have a device that had a partially functioning screen and I spent a lot of time rotating the phone and using this!

Within Settings > Developer Options is some options that will help you diagnose the problem, such as Show Pointer Location, or perhaps Show Touches. Or you could create a small app that simply displayed touch input.