I thought I just had a great idea but sadly I'm still at a dead-end.
The idea was to use "Show region" on the User Level part of the Profile Update page (if Admin), then change the access rule to "Logged in as visitor". Sheer brilliance!
But sadly, although this now allows any visitor to view and update their profile (except their user level - good), and it allows Admin to log into any Profile Update page using the user's log-in (good), Admin can still only see User Level for their own Profile. D'oh!
So anyone who can tell me how to use the Profile Update page to update profiles gets a gold star :-)