If were still on a gold standard I would favor keeping it, but I don't think we are giving up fiat currency. In that case, the monetary system will be subject to political influence, although central bank independence is supposed to reduce that influence.
You’re right. It is too late to fix. The problem lay in the creation of the Fed itself and the lack of realization that its board members would all become political hacks.
As it is the only thing that keeps the fiat money system from collapsing is that the rest of the world simply can’t risk calling it into question. Collapsing the US economy would send the rest of the planet back to the Dark Ages.