It's a fair question to ask "who are independent executive agency heads accountable to" in a constitutional context. It is true that the Executive Branch has grown far beyond what the Founding Fathers could have imagined, but the idea of a unitary executive is that the President is responsible and accountable for everything that happens in the Executive Branch. If the voters don't like what the Executive Branch is doing, they can replace the President in the next election. What happens if voters don't like what independent executive agencies are doing? There's no democratic recourse.

Think of a scenario where a President was elected with a large-ish majority and promised during the campaign to change broadband regulations to reduce broadband prices across the country. Unfortunately, the FCC commissioners were all appointed by the previous president and block this policy change that the voters clearly support. How does that square with democratic accountability?