One could argue that Hamilton's elected president for life is little more than a dictator.

Hamilton's President for Life still needs the Congress to enact laws, appoint the Supreme Court and pass the budget. Unless he chooses to rule by executive order. And even then he can be impeached.

When the capital development of a country becomes a by-product of the activities of a casino, the job is likely to be ill-done. — John M. Keynes

More importantly, in their political culture, was the question of who to elect phrased as "to whom shall we delegate our sovereign power?" or was is phrased as "who do we think has the true divine mandate?"  In the former sense, the source of authority is the people themselves; in the later, the source of authority is God.  So an elected monarch would still be very different than an elected president for life.

When I write divine mandate I am thinking of the Asian concept that the ruler rules with the mandate of heaven.

A broader example: In traditional Islam, the caliph was selected by consultation.  This was a form of election, it varied between election by consensus, and election by majority.  Of course, "the people" meant the Community of Believers (Ummah) which excluded women and non-Muslims.  But most importantly for this discussion, they weren't delegating their power to the caliph; all power and legitimacy came from Allah and was delegated by Allah directly to Muhammad.  The Ummah was merely using the "wisdom of the crowd" to decide who was Muhammad's true successor.  In that sense, the caliph could be said to be a sort of elected monarch.

Do you see the difference?

So aside from snarking Bush as some sort of theocrat, which is absurd, I just don't see the utility of the term elected monarch to describe the US Presidency, or even Hamilton's ideas.

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I am the most conservative Unitarian-Universalist you will ever meet.