The party chair decides who the party invests in, rules for the debates, suggests primates and orders.
But... Two things have altered that
1. Citizens united....simply put the biggest mega donors run the party by buying the most seats, especially given the gerrymandering of almost 90 percent of house districts. This is why the Kich brothers always invested in the legislature rather than presidential elections. Think it is easier to, and sure fire wins, to buy 100 representatives and 10 senators, than maybe buy a president. See Joni Ernst in the Senate and the coup caucus. The Koch's in many ways ran the party by buying enough politicians to stop (that filibuster thing that Moscow mitch really exploded) any legislation that they didn't want. Again, using Ernst as an example. Iowa's biggest industries after agriculture are wind turbines and biodiesel. She has voted against both repeatedly because they would hurt the Koch empire.
2. The cult of TFG...the Koch model works when the politicians you bought are afraid of you supporting someone else in the primary. Now. The 'base' for primaries are the MAGA loons who will oust traditional or even AstroTurf rethugs bought and sold by billionaires because mango Mussolini called them rino.
Because of #2, the RNC and it's head which should be running the party at the behest of the billionaire donors who get people elected are now doing things because mango Mussolini, who gets people best in the general election, wants it.
In the past, I would have said the money behind the federalist society, the Koch brothers, etc. were really running the Republican party thru the RNC...now it is a cult