Isn't having elected representatives control government agencies more democratic?

No, the same way it isn't more democratic if all branches of government all tie the same party line.

Modern democracy is based on the separation of powers and, in the vast majority of countries, a constitution or charter(s) that can't be overturned by a simple majority vote. For example, a party can't suspend Habeas Corpus on a whim, even if said party won the presidency, the Congress and the Senate.

But just ask the Supreme Court who carved an exception for the FED and not other government agencies...