It's been awhile since I've done much reading on the French Revolution, but perusing through some older answers on similar topics leads me to a couple key questions. First, from my understanding, at least part of the downfall of the Ancien Regime could be attributed to the horrible state of the crown and states finances in the later half of the 18th century. While a few posters mentioned the loss of French North American colonial possessions, no one seems to draw any connection to funding a costly intervention into the American Revolution. Second, from an intellectual standpoint, how well known were American Political thinkers within French Revolutionary circles? Obviously there's Lafayette, the Declaration of the Rights of Man and the Jeffersonian connection, but how well known were the founding fathers and their (admittedly diverse) philosophies in France?
I know I asked a few pretty big questions that technically go into different areas, but both contribute to my general question of the affect the American Revolution had on the downfall of Bourbon France?
French involvement in the American revolutionary war had a direct impact on the downfall of the Old Regime. It did so in two key ways: Bankruptcy and ideology.
Bankruptcy
Firstly, as you noted, participation helped to bankrupt the nation. A significant cost was incurred due to the construction of a fleet which would enable France to ferry goods and men to the North American theatre as well as challenge Britain's naval supremacy. As Historian Peter Kroptin (1909) noted, “It might be said also that the war in America, during which France had to build an entire fleet to oppose England’s, completed the financial ruin of the old régime and hastened its downfall.”
The debt incurred from the war, combined with the debt incurred from previous conflicts, triggered the financial crisis of 1787. This crisis facilitated an aristocratic revolt (initially led by the Assembling of Notables, then the Parlements) in which the nobles refused to pass financial reforms aimed at addressing the bankruptcy. They did this to force the summoning of an Estates-General, which the nobility would control if it sat in its traditional format. The aristocratic revolt gave way to a general revolt in the wake of the Day of Tiles in Grenoble and the Vizille Assembly in the middle of 1788. This revolt forced the government to call the Estates-General, the body which would transform itself into the National Assembly.
Without an impending bankruptcy, the government would never have had to contend with the aristocratic bodies which conducted the aristocratic revolt against Bourbon absolutism. Furthermore, it would not have had to contend with the general revolt of the populace and its calls for the establishment of an Estates-General and the constitutional monarchy it implied. While the Old Regime had multiple structural issues and was in need of reform, the revolution of 1789 in the form we know it would not have occurred without French involvement in the American Revolutionary War (or a similar conflict). No conflict means no (additional) debt, and thus no bankruptcy to avoid (at least in the mid-1780s). Without an impending bankruptcy, there would have been no need to summon an Estates-General. As the Controller-General of Finances Turgot warned while debating the merits of entering the conflict, "the first gunshot will drive the state to bankruptcy." (Doyle, 2018).
Ideology
How the conflict impacted France from an ideological/intellectual standpoint is more difficult to ascertain, but there is little doubt that it had a impact. Even before the conflict, the potential for French intervention to have unintended consequences was known to the French court. According to Historian Simon Schama (1990), the Queen Marie Antoinette warned that a monarchy should be wary of aiding republican rebels against another monarch. The Queens concerns were not unwarranted.
French intervention contributed to the success of the American revolutionary experiment. As the experiment continued to work, America offered a viable and visible alternative to Old Regime France (and feudal Europe more broadly). Prior to America, the ideas propagated by the Enlightenment (natural rights, liberty, equality etc.) had not been used as the foundation for an entire society (particularly one covering a non-insignificant region and population). With America's success, France (and Europe) witnessed an alternative way of life which could be emulated. As Historian Richard Mackey (1976) wrote, “The American Revolution produced for the people of France an educational program of gigantic proportions”, and that, “The French intellectuals could foresee in America the Nirvana of the future golden age and a model for a new France.” The impact of this educational program can be seen in the Declaration of the Rights of Man (which was first proposed by Lafayette and influenced by Jefferson (Lafayette's version of the Declaration was not the one eventually adopted however)).
The model for a new France which America provided not only energised key members of the French intelligentsia (including veterans such as Lafayette) to pursue reform, but also weakened the support for the existing regime. Furthermore, this deterioration of support occurred not only amongst the liberal nobility and the intelligentsia but amongst members of the Third Estate as well. Historian Robert R. Palmer (1959) notes, “Whether fantastically idealized or seen in a factual way, whether as mirage or as reality, America made Europe seem unsatisfactory to many people of the middle and lower classes, and to those of the upper classes who wished them well. It made a good many Europeans feel sorry for themselves, and induced a kind of spiritual flight from the Old Regime.”
It is noteworthy however that while the success of America's experiment with enlightenment teachings influenced the revolutionaries, it did not dominate their decisions. The French rejected both Lafayette's version of the Declaration of the Rights of Man and bicameralism in creating their own foundational/constitutional structures. Furthermore, even after France became a republic, the two constitutional systems remained distinctly unique. Thus, while America influenced the thinking of the revolutionaries and weakened support for the Old Regime, its influence of the french was limited and did not facilitate a wholesale replication of American ideas across the channel.
Hope that helps to answer some of your questions.
The above points, and other discussions surrounding French involvement in the American Revolutionary War and its consequences, can be found in my podcast 'Grey History: The French Revolution'.