Obama picked Schultz. I think you may be confused.
No, I'm not the least bit confused. You are.
Obama was incredibly popular within the Democrat Party itself. So, when he suggested DWS for head of the DNC, the DNC obliged by electing her. And that has been the custom within both parties. If the President wants someone to head up the party, the respective national committee goes along with his wishes and elects his candidate. But only the committee itself has that actual authority.
What neither you nor Trump apparently understands is that is a custom only. It is simpy a function of the usual respect the President shows for his party, and the recognition of how important his support is to the party. But the President has no legal power to replace the head of the National Committee on his own authority.
Trump has essentialy been at war with the RNC itself. Stupidly, I might add, because most of the things of which he has complained are controlled by state GOP committees rather than the RNC. In any case, both Trump and the majority of his supporters (at least here on FR) have been openly disdainful/insulting towards the RNC, and towards the party as a whole. Heck, a large number of you guys apparently believe in "uniparty" and so openly want to destroy the current RNC. Why in the world would members of that RNC cooperate in their own destruction?
So I think it will be a cold day in hell before the people he (and you guys) have been insulting install as head of the RNC someone he prefers, unless he actually makes amends on his end.
As of right now, I expect the majority of the RNC would just flip the proverbial bird towards any Trump demand to replace the head of the RNC.