On your list of Saved Searches, switch the "Show All Private Searches" option to Yes instead of -All-. You'd think that the -All- option would show more searches than the Yes option. You'd be wrong. 🫠
One of those bugs that's been around forever and that will likely never be fixed.
Anyway, switching that to Yes should allow an Admin to see any search in the system. I think that a search owned by another Admin (and not public) will not show up in Global Search, but it should show up on the list of searches if you switch that option to Yes.