They aren't 'independent' they are 'a mix between executive and legislative'. The Supreme Court decisions are Meyers v US and Hunters Executor v US. And I'm not a constitutional scholar but my reading of it is that the protections in question come from the legislative delegating some of their power to the executive, think legislative actions (researching laws, etc) but retaining their constitutional prerogative to protect them from executive control.
This is something that has existed for a very long time but has been changing lately and will almost certainly show up in the Supreme Court again.
“ Myers v. United States, 272 U.S. 52 (1926), was a United States Supreme Court decision ruling that the President has the exclusive power to remove executive branch officials, and does not need the approval of the Senate or any other legislative body.”
This is something that has existed for a very long time but has been changing lately and will almost certainly show up in the Supreme Court again.