> Who would they be accountable to if not the US President?

Accountability does not require absolute control. A subordinate official can be permitted (even mandated) to exercise independent judgement and still be accountable for mis-, mal-, and nonfeasance to a higher authority.

And this has, with different precise parameters, long been the statutory model governing the President’s relations with much of the executive branch, with different specific rules applicable to civil service employees generally, Inspectors-General and a few other specially-designated employees in regular departments, and independent agencies.