To a point. I have found that subordinates always have to feel that they have a "little secret edge" over their managers.

In my experience, this is both harmless and necessary. If I, as a manager, can't handle that, then I'm kinda screwed.

But it is also highly dependent on the types of people we manage.

Source: Manager for over 25 years, of a bunch of folks much smarter and more experienced than me.

You can lose the respect of your subordinates, but that's different than losing face.

Although one may lead to the other, they are different things.

I think the OP meant: in private, face is only lost when criticised by a peer or superior.

If your superior is managing up well, then it truly doesn’t matter if subordinates lose respect. Probably laying groundwork for new ones well in advance of that becoming a real issue for their status.