Here’s something that you can already do. It’s not quite as convenient as a hotkey, but just as effective. Go into Preferences - Display - Name and select the “Given” format, so that only first names are displayed. If that’s not enough privacy, then simply add a new name display format that consists only of static text such as “(private)”.
There may be other things that need to be masked in screen shots, such as source or media titles like “Birth Certificate - John Smith”.