Woah that is weird.
One additional thing that may be helpful. I have “Load UI” turned off in my Blender user preferences.

and I just downloaded the start.blend file and everything seems to work fine for me because I kept my standard UI settings (I’m even using blender official 3.4.1 and it still works).
I can see the edit mode stuff in the starting file just fine (note: I did do an unwrap already when I took this picture but that was all I did)

If you want to do similar. I recommend saving your work, closing the file. Opening a fresh blend file and navigating to user preferences and turning that Load UI off.
I personally don’t like my UI changing to whatever someone else’s is when I open their file.