Hmmm... I don't think it is really anything like monarchy. The latter is a constitutional system which forms the foundation of a couple dozen of the freest and most successful states in the world. The former is a type of party which is held in a tiny number of social circles. I'm not sure what a debutante ball is working to do other than allow a tiny number of wealthy people to declare their own exclusivity, and I am certain that no portion of the public which could be fairly described as "the people" even know about it, much less want it.
That said, while I agree these balls are anachronistic, if the participants enjoy them, I see no reason for them not to continue to have them.

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Isn't this sort of thing getting a bit outdated now?
The same thing could be said about monarchies. These debutante balls serve a purpose in their social circles. It's not outdated if it works and the people want it