I think you are right that the US constitution, as originally written, doesn't provide any internal "checks and balances" on the executive branch, other than the President. Congress and the judiciary act as external checks and balances on the President (and also inferior officials, since Congress can impeach inferior officials, and the courts can rule against them). The President acts as an internal check and balance on the executive branch (powers to fire inferior officials, direct them, demand information from them)

Not to say that I think this good policy or constitution design – it grants the President an essentially monarchical position. As The Knoxville Journal once said (9 February 1896), "Great Britain is a republic with a hereditary president, while the United States is a monarchy with an elective king". I think the more collegial form of executive branch leadership found in the Westminster system – in which a Prime Minister has to continually keep the confidence of their party, since they can be removed at any time for any reason (no allegations of misconduct required); in which Cabinet makes decisions by majority vote (and the PM sometimes loses the vote), unlike the US Cabinet where no votes are taken – leads to better governance.

Maybe, one day, "Prime Minister of the United States" will be a real job title