In The English Constitution, Bagehot makes a case for the superiority of the British cabinet system over the American. When I read this, I was not really convinced, and may not be now. However, it appears to me that as the federal government has expanded--as the cabinet has expanded, and one might say diffused--the cabinet has lost power.
The third through fifth presidents of the United States had served in the Cabinet, and Van Buren and Buchanan after them. That I can recall, the next member of a Cabinet to become president was Herbert Hoover. The Roosevelts topped out at sub-cabinet level, as Assistant Secretaries of the Navy. George H. W. Bush was director of the CIA, but long before that post came to be part of the Cabinet.
Since the earliest days of the republic, there have been complaints of the president ignoring the Cabinet and taking advice elsewhere. Henry Adams quoted John Randolph in a speech of 1806:
The first question I asked when I saw the gentleman's resolution was, Is this a measure of the Cabinet? Not of an open declared Cabinet, but of an invisible, inscrutable, unconstitutional Cabinet, without responsibility, unknown to the Constitution. I speak of back-stairs influence-of men who bring messages to this House, which although they do not appear in the Journals, govern its decisions. Sir, the first question that I asked on the subject of British relations was, What is the opinion of the Cabinet; what measures will they recommend to Congress?--well knowing that whatever measures we might take they must execute them, and therefore that we should have their opinion on the subject. My answer was (and from a Cabinet minister, too), 'There is no longer any Cabinet!'
(The Cabinet then included James Madison and Albert Gallatin; but Randolph disliked Madison, and may have been disenchanted with Gallatin as impossible to intimidate. In any case, Randolph's quarrel had to do with the President as much as the Cabinet.) I remember accounts fifty years ago of Richard Nixon ignoring his Secretary of State in favor of Henry Kissinger, then his National Security Advisor. Have matters changed at all?
It is said that the President elect's last cabinet considered declaring him unable to continue under the fourth section of the Twenty-Fifth Amendment, and did not. It is also said that the current President's staff kept the Cabinet away from the President and unable to judge his state. Perhaps Bagehot was right.