Sorry about the very late reply! Life has been quite busy the last few weeks, and unfortunately refuses to schedule itself around good Reddit questions.

So, one thing that makes your question a little tricky to answer is that rather than being a question of "fact" (ex. What was the Allies plan if the Soviets surrendered?) it is a question of judgement by historians. There is no objective answer to what event was the beginning of the Modern Age" because there isn't an objective Modern Age (and certainly not one event that objectively starts it), its just a helpful framework for us to view history through. There certainly are individuals (scholars or not) who view the American Revolution as the beginning of the modern age, and they aren't necessarily wrong, they just prioritize different things and have come to a different conclusion. There are plenty more events based on the same Enlightenment principles as the American and French revolutions that you could argue are the real beginning of the Modern Age - the Dutch Revolution, the Haitian Revolution, the Revolutions of 1848 - but the French revolution is just the singular event that most historians have (sort-of arbitrarily) set as that point.

Another thing to keep in mind is that for a very long time the discipline of history was (and to some extent still is) very Eurocentric. Events that affected Europe were/are often given more importance than equally large events that affected other parts of the world (at least in English-language historiography). If we examine direct, short term consequences of these two revolutions, we see that the American Revolution... didn't really change much about the world order, certainly not the European order. Britain was still a dominant imperial power, and there was little fighting on the European continent itself. The United States wouldn't be seen as a legitimate equal in geopolitics for decades, at least. We can argue about how much the French were inspired or influenced by the success of the Americans, but ultimately most Europeans would have discussed the American Revolution as a faraway experiment in Enlightenment thought that may or may not work and probably won't have much influence on Europe or the rest of the world. The French Revolution, however, led to a decade of war that raged across the entire continent, destroyed the Holy Roman Empire that had been dominant in Central European politics for centuries, popularized the metric system, and set the stage for the Concert of Europe and balance-of-power politics that would define European diplomacy for decades. We can argue about long-term impact - the French Revolution is credited with igniting the nationalism that would drive German and Italian Unification in the later 19th century, both revolutions inspired or influenced independence movements in Latin America, and of course the United States would later become a global superpower - but for immediate European impact that, for a long time, historians would care much more about, the French Revolution wins out.

Finally, there are some legitimate differences between the two revolutions that, depending on how you think the Modern Age is distinct from previous eras, could help separate the two. Principally, the American Revolution was primarily a local rebellion for national independence, keeping a lot of institutions intact, while the French Revolution was a complete social revolution that completely reformed the government and culture of France. You'll notice that most of the Founding Fathers in America were already quite wealthy and influential in the colonies before independence, as were the majority of early American legislators and politicians. (I think this factor is a little exaggerated by popular history, but that is neither here nor there.) American society was largely the same as before the Revolution, just with the upper crust of colonial British control cut off. Rural planters were still the elites, with cities rapidly growing but their influence tempered by the Senate and Electoral College. For most Americans, who lived in semi-autonomous colonies with (limited) democracy thousands of miles from London, no longer being subjects of a king didn't really impact them. The French Revolution, on the other hand, turned everything on its head. Famously, many members of the aristocracy (and lots and lots of commoners) were put to the guillotine, with the king himself being executed, which of course wasn't even a remote possibility with the American revolution. Many revolutionaries were members of the urban working class, which was a relatively new development in the West. The organization of the French government was completely restructured; the calendar was replaced with a new one developed by the revolutionaries; the Catholic Church (and religion in general) were targeted, with the goal of eventual replacement; France even tried universal (male) suffrage in 1792! So, if you are defining the Modern Age as the era of populism, democracy, and the creation/rise of an urban working class and bourgeoise, then the French Revolution is certainly a better starting point than American independence.