The US constitution lays out three AND ONLY three branches of government. The congress cannot create a fourth without an amendment. If they create an agency in the executive branch, by definition it reports to the head of the executive.
Sure. So explain the results of Trump v. Cook, which involve exactly that.
The same justice, on the same day, issued one opinion that says Congress can't put limits on firing FTC chairs, and another that Congress can put limits on firing Fed board members.
The ruling in Slaughter was not that president can fire commissioners no matter what branch they are in. It was never established in Trump v Cook that the federal reserve lies in the executive branch.
Your attempt here to falsely portray an inconsistency that doesn't exist. It's a different question as to whether the federal reserve is in the executive branch. You'll have to show the federal reserve is in the executive branch if you want the same ruling to apply or claim this inconsistency.
But that's insane, right?
"You can make an agency the President has no control over, but you can't make one the President has some control over."
It would be pretty nuts if the president could just fire whomever he wants no matter what branch they are in based on a simple declaration he considers them to be in the executive branch.