I’m trying to do the same. Aaron’s post, provided by MichaelH, doesn’t remove functions but adds additional features.
So I sought advice at these three helpful articles:
No success.
Admittedly I’m not a pro at this. But it looks to me like the Twentyten setup functions and actions are all wrapped together in a way that makes them hard to pull apart and remove. For now I’ve made my own copy of Twenty Ten, renamed it (with due credits to the WordPress Team), and have simply removed the actions I don’t want.
But if someone can successfully demonstrate how to do this in a child theme — i.e., remove just the custom header actions — I would certainly appreciate the help as well.