This is a very interesting notion, I am from the UK but have a personal (and academic) interest in the American Presidency, so alongside extensive reading, I feel personally able to assist you in understanding this.

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The notion that the United States is an elective monarchy no doubt stems from the perceived individual power of the office of POTUS. If we consider the number of traditions associated with the Presidency, such as the playing of Hail to the Chief, the inauguration ceremonies, State Funerals where the casket lies under the rotunda and a service is held in the National Cathedral, the Presidency has become almost as ritualistic as a traditional monarchy. As a Brit myself, I couldn't help but be reminded of a ceremony like the Trooping of the Colour when I watched the coverage of President Bush's funeral last November. Queen Elizabeth II has Buckingham Palace, Windsor Castle and Balmoral (amongst others), he President has the White House, with the Oval Office and the Resolute Desk and Camp David, which are just as significant symbolically. He also has a title, 'Mr. President,' and features on currency, just like the British Monarch, and even has his own day, President's Day on the third Monday in February.

Interestingly enough, in the aftermath of the Revolutionary War, the idea of having a Monarchy in the United States was debated. Supporters of the idea wished to have George Washington installed as the first King, but this did not occur, obviously. Washington is revered in the public consciousness still, his face on currency, the capital bearing his name and indeed, a magnificent monument on the National Mall being constructed in his honour (it was supposed to be more extravagant, but these plans were halted). There were also plans to bury Washington at the centre of the Capitol, but these never came to fruition, with the objection of his wife. Thus, the public veneration Washington in particular is clear, alongside select other Presidents including Lincoln, Jefferson and FDR who have received their own monuments in DC. (A book on this subject would be Kathleen Bartolini-Tuazon's "George Washington and the Presidential Title Controversy of 1789: For Fear of an Elective King.")

I'm not sure I would agree, personally, on the concept of the U.S. being a Monarchy. In the eyes of the public, perhaps, yet from a political science perspective, the real 'power,' lies with the US Congress, the President can only pass Executive Orders by himself, requiring Congress to do much more, including authorise wars. (A good book on this debate is The Imperial Presidency by Arthur M. Schlesinger or the Presidency and the Constitution by Genovese and Spitzer).

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The United Kingdom, by comparison, despite the attempts of the various Reform Acts from the Victorian and Edwardian Era, has a Parliament comprised of a relatively small slice of society. In the Georgian era, there would be little choice for constituents and only wealthy men were able to vote, keeping the 'power', i.e. the seat, amongst themselves, filling Parliament with their 'type.' This undoubtedly continues, the vast majority of MPs having gone to private schools and elite Universities such as Oxford and Cambridge. Jeremy Corbyn, the current leader of the Labour Party, would be an exception to this, but he does not get favourable treatment from the press. Furthermore, before attempted reforms, seats would pass from father to son, keeping the power in bloodlines. For example, current politicians such as Jacob Rees-Mogg have fathers who were MPs, and David Cameron was a descendant of the prominent Astor family. No doubt this has opened up in recent years, with the first black female MP Diane Abbott still serving, and increasing diversity with individuals such as Sajid Javid or Chukka Umuna. But the Parliament remains disproportionately white and male.

If we consider more recent British Prime Ministers, although Margaret Thatcher was in office for an extensive period of 11 years from 1979-1990, she attained only 44% of the popular vote at her peak, due to the peculiarities of our electoral system, with its 600+ constituencies and not voting for the individual, but rather, a local MP. Much akin to the frustrations of the electoral college.

With the examples I've just highlighted, there are clear aspects of a hereditary British Parliament, so no doubt the argument could be made for a hereditary republic existing.

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I can't, nor would any other Historian, be able to say whether this is 'true': it's a historiographical debate, and one that will probably never be resolved. But I hope what I have just written assists your understanding and helps you form your own conclusion.