The Glorious Revolution and an American Monarchy
Currently, while the House of Windsor is considered sovereign, that sovereignty is within the limits imposed by Parliament under the Glorious Revolution, which cemented Protestant rule in England, thus ending the religious civil wars of that era with both the Catholic James II and the Roundheads. The Hanoverians were thus recruited to restore the monarchy.
These arrangements are by tradition rather than a written constitution. They include the retention of royal properties, an annual appropriation to maintain the Royal family, titular head of state both Kingdom and Commonwealth and the permission of the Prime Minister to form a government that is actually elected by the people of partisan majorities or coalitions. The Monarch is then given the responsibility of endorsing the program of the majority in the Queen's Speech.
I find that last bit objectionable. A true sovereign should have the right to designate the actions by the views of the ruling party as such rather than as Her Majesty's Government. Indeed, the Prime Minister should speak for his or herself. Any remarks by Her Majesty must hers alone, including arguments against any actions of the majority which violate the rights of the citizens of the realm. She is also the secular head of the Church of England and Defender of the Faith, the last bit being a Papal grant that was retained as a title.
In former times, the Monarchy ruled with the sanction of the Church. Before the rights of the nobility, and later the people, were laid out in Magna Carta, the Catholic Church was the main check on the exercise of royal power. This had been true since the conversion of Clovis and subsequent coronation. In ancient times, monarchs served as gods in the name of the local pantheon, although the period of democracy in Athens and in the Roman Republic, as well as the revolution due to the invasion of the Sea Peoples, which ended the Bronze Age overthrew the god-kings from all nations but Egypt.
The Glorious Revolution displaced the Church as the arbiter of sovereignty, thus ending the proclivity of kings to make war and sects to declare revolutions in the name of religion. Even then, and until recent actions having to do with the descendants of Prince William, Catholics were denied the dynastic right to rule. The implications of this for the title of Defender of the Faith have not been considered.
I would argue that the constitutional legitimacy of the monarchy as head of State (but not government) should be vested in the Anglican Church, not the representatives of the people. The end result would be the same, except in bad behavior, where monarchs could be deposed.
This theory of royal sovereignty allows the creation of an American monarchy. While the Constitution does not allow the issuance of royal and noble titles or the provision of subsidies to the Crown, royalty could still be conferred by the Episcopal Church. Unification with the Methodist and Presbyterian Churches would add legitimacy. Greater still would be unification including an American Catholic Patriarch, provided the bishops were appointed locally rather than being appointed by the Holy See.
Why bring this up? Prince Harry has become an American resident married to an American citizen whose child is entitled to dual citizenship as well. In essence, we have our own royal. The COVID concert broadcasted last night included his participation. He has become our American Prince. The Episcopal Archbishop of Washington would be within his rights to crown him King under the theory outlined above. Indeed, he may be more popular here than the House of Windsor is back home.