With the Declaration of Independence, these colonies declared "That these United Colonies are, and of Right ought to be Free and Independent States" (each on equal footing with the state of England, the state of France, the state of Spain, etc.).
The Constitutional Convention created a union (a compact, or contract) among these sovereign free and independent states, with specific enumerated powers delegated to that central created entity (and most of those delegated tasks pertaining to foreign affairs). The federal government is not a sovereign entity. The states and the people are.
Therefore, the whole discussion about royal bloodlines in the federal government is moot.
Believe what you will. We can all sit here and discuss how things “should be”. Doesn’t matter what the theory is when reality is very different. One thing is certain, precedent has been set for over 200 years. And we shouldn’t expect anything different now, or in the near future.