Perhaps you're getting mixed up on the nomenclature? There are "independent agencies" in the US Executive Branch, but these aren't independent of the entire US

Perhaps you're getting mixed up on the nomenclature? There are "independent agencies" in the US Executive Branch, but these aren't independent of the entire US Government; they're just not part of a single Department. For example, NASA could have been part of the Department of Transportation (and thus report to the Secretary of Transportation), but it isn't and reports directly to the President. Agencies are typically run by someone appointed by the President and confirmed by the Senate, but the agency heads are typically supposed to be less political than the cabinet. For example, the President can fire the Secretary of State on a whim, and might do so to indicate a policy shift. On the other hand, the members of the Nuclear Regulatory Commission, for example, are appointed for fixed five year terms and shouldn't be removed for political reasons. Regardless though, they are still part of the government.