The whole point of the trillion dollar coin idea was that it allowed the executive (aka the president) to create money without asking the Fed (and thus by pass the congressional deb-ceiling debate). This only illustrates the point that president doesn't directly control the Fed - he does appoint the chairperson (Bernanke, Greenspan et al) but he cannot order them to print money. The treasury however does have responsibility for minting actual coins, usually treated as a minor housekeping type matter (because the vast bulk of money is not actual physical coins and notes, of course) and even then only under congressional mandate.

In the case of the trillion dollar coin they had passed legislation that the Treasury was to be given the leeway to print a new platinum coin - if they wished, and at their discretion as to its denomination.

All of which is to say that this whole trillion dollar coin thing only emphasizes the extent to which the president does not control the Fed.

FWIW this is very common in western democracies. Even when the central bank is not a privately owned institution it is almost always managed in a very hands-off way, ostensibly (though perhaps not practically) to stop short term administrations from playing with things in a way that might affect long term currency stability.