To begin with, the leading figures in the early United States definitely did care about the French Revolution. It was an extremely important topic for politics in the early republic. Your point about the monarchy that supported the U.S. being overthrown is a perceptive one, but remember that during the French Revolution's early stages, many observers felt that a Constitutional Monarchy would be the likely end result of the whole thing, in some ways much more like England's Glorious Revolution than America's revolution. Additionally, many people made the argument that America's debt was to the French people, rather than to the King, who had acted in his own imperial interest.

As for the "Founding Fathers," many of them followed the basic trajectory of the majority of Americans (which was something like rapturous enthusiasm in 1789, some doubts creeping in in 1793, and definite opposition and paranoia beginning in 1798). There were two major exceptions to this: Thomas Jefferson and John Adams. Adams was skeptical about the French Revolution pretty much from its onset. He was more or less alone in this opinion, especially in the early 1790s. His 1790 book Discourses on Davila made the argument, in part, that the French Revolution would reprise the French Wars of Religion from centuries past. This was an abstruse text that wasn't terribly widely read. Adams' cynical view of the French Revolutionaries continued throughout the decade.

Jefferson, on the other hand, was arguably one of the Revolution's biggest proponents in American political life. Even after most Americans turned against it, Jefferson was totally unwilling to give up on it. One reason for this is that he had actually lived in France during the revolution's earliest and most hopeful stage. After he returned to the United States, one of his responses to Discourses on Davila was to help Tom Paine's Rights of Man be published. Even the death of Louis XVI, which as you suggest out sparked controversy among many Americans, didn't faze Jefferson. In 1793, he wrote to James Madison: ""The death of the King of France has not produced as open condemnation from the monocrats as I expected… It is certain that the ladies of this city, of the first circle, are open-mouthed against the murderers of a sovereign, and they generally speak those sentiments which the more cautious husband smothers."

However, as the revolution's body count mounted and France ran into diplomatic problems with the United States, Jefferson and his allies (here I'm including Madison) became less vocal about their Francophilia, even as Adams and his ilk (here I'm including Hamilton, which he probably would have resented) stepped up their criticism. Federalists increasingly looked to Edmund Burke's Reflections on the French Revolution. Additionally, Hamilton and his friends attempted to move the young nation out of France's orbit towards England, which they viewed as a more suitable trading partner. Eventually, even as Jefferson and some of his friends privately noted their continued optimism about France (In 1796 Jefferson wrote, "This I hope will be the age of experiments in government.") most of the "Founding Father" figures had essentially soured on the French Revolution by the end of the 1790s.

Note: This topic is part of what my dissertation is based around, so I could keep writing about this all day, but I don't want to over-answer your question. Feel free to ask for more if you're interested, or for sources for particular questions.

Sources: Conor Cruise O'Brien, The Long Affair: Thomas Jefferson and the French Revolution; Louis Sears George Washington and the French Revolution; Philipp Ziesche Cosmopolitan Patriots; Rachel Hope Cleves The Reign of Terror in America; Seth Cotlar, Tom Paine's America