Maybe I'm old timey (like 2005 old timey), but don't they hold elections for these kinds of things?
From the point of view of CNN the chair of the RNC is an appointed position. From the view of one who read's the RNC's rules, it is an elected position and the President does not have a vote.
So, I ask Kos and I ask Redstate (and here).
Ironically, Kos is happier than Redstate about all of this.