To what extent is a coincidence that the Dec. of the Rights of Man and the Citizen and the USA Dec. of Independence were approved so close in time?

Well played sir, well played! Drake has persuaded me to offer a bad answer. This is a bad answer because I'm going to cite a set of generalizations without either the sources that I prefer or the extensive scholarship of the god king of H:SE.

If you want a one sentence answer, growing communications, a broader distribution of economic power to individuals, and an increasingly complex world demanded a more democratic distribution of power and governance, and a new social contract that supported the resulting society.

If you want to pick apart some of the underlying factors:

1. The American revolution did not emerge unheralded; it was the outgrowth of prior actions & trends. The Glorious Revolution in Great Britain established the primacy of Parliament and settled many of the issues that were raised in the Protectorate. The Dutch already had a Republic; the Swiss already had communes. There were antecedents.

2. One of the fundamental underpinnings was economic. Others on this forum will disagree with all the details of my economic analysis, but I think there is general agreement that national economic systems were all broken. The Stop of the exchequer, France's problems with Necker, Spain's recurrent bankruptcy are all symptoms of the same problem. Without going too deeply into the problem, I'll merely assert that nation-states needed lots of money to fund a strong executive and a strong national defense. The world international economic system was mercantilism, which provides very few opportunities for nation states to cooperate, and makes competition exceedingly expensive. France has more serious problems because their economy involves less novel commerce and more traditional landholding - France's economic system are deeply coupled with governance, tradition, and other systems that are highly and effectively resistant to reform. (Taxation and French budgeting are the crux here)

3. Partly related to the above is a growing problem with governance (how nations exercise executive power), and the principal-agent problem. Kings had huge authority, but they had to exercise that authority through agents (the term "placemen" is used alot in US revolutionary arguments). The complexity of society had grown (again, here I'm waving my hand; specific details will be hotly debated, but I think the general conclusion is acceptable). There was more economic activity, involving more participants. Everything that a government must do was more complicated. Commerce, defense, law enforcement, regulation, civil rights, immigration, customs, etc. Government had to add new complexities that had not been anticipated. Printing, health, and most of all colonies. France lacks any effective representative function - effectively all power is executive and deeply coupled with landholding.

4. Governance again - In order to address the growing complexity of governance, including most especially the rising wealth and power of what was known as "the middling sort" and would eventually be known as the middle class, rights were granted to chartered organizations - most notably Parliament in England. England after the Glorious Revolution was still trying to figure out the concept of a loyal opposition and how a pseudo-democratic institution could work with the King. Everyone acknowledged that Parliament sucked at executive power, but nobody wanted to reinforce monarchy. In the old system the King appointed someone to handle the problem and held that individual accountable (removal of position, or death) if the problem didn't get solved. That doesn't work in a parliamentary system for complicated reasons that eventually get summarized as public choice economics Once again France's ancien regime lacks a representative function. An accident of French history permitted two very effective kings to defer these problems until they came to rest in the hands of an individual who was fundamentally unable to handle the issues, and had no effective advice, counsel or assistance. I really should include a paragraph on the growing power of the nation state, but if I start that paragraph, I'll never stop writing.

5. Colonialism. Colonies complicate governance and support the economic dysfunctions of mercatilistic competition. Each nation did colonialism different. It is well worth listening to Professor Rakove's lectures on the topic. Spain sent landlords who owned the native population and had no intent of staying. France taught the natives that they were now Frenchman and their heritage was now Roland. England displaced the natives and sent people to own land. But land ownership in England carried certain rights and prerogatives that were not available to those who owned land in America. France lost the first imperial war but France's inability to delegate power effectively from the Monarch further crippled its ability to manage colonies. I suspect, although I can't prove, that the most expensive empire is the second best. Once again, I'm not good enough to explore that concept so I'll just hang it out there. Aside: I will draw your attention to @Steve Bird's comment above "They may not have been ultimately successful, but the French certainly put a whole lot of effort into trying to form colonies there" [North America] - which I think is relevant.

6. Factionalism - this is a deep and complex subject, but it is worth noting that for reasons that are partly due to religious factionalism, many of the people who moved to America were politically aligned with Whigs. For simplicity, we'll identify Whigs with classical liberals and note that they're going to be more comfortable exploring alternative forms of political power. That isn't accurate, but I'm already running on too long. France has a different problem- they don't have factionalism, they have a version of the 1% problem - Wealth and power are increasingly coupled and concentrated in an ever narrowing set of people. They don't have a Whig problem, their problem is that the legitimacy of their government is failing; the government serves the needs of a handful of people who are structurally unassailable; everyone outside that handful is by definition more Whiggish than the rulers.

7. Religion - another book length answer and particularly important in France. Religion was an institution that had grown to be tightly coupled with the traditional exercise of power. Religion lacked the ability to change and adapt to the more complex world. In France the majority of religious institutional members (local priests) aligned with the completely disenfranchised common people; the minority of the religious establishment (Archbishops and above) controlled all the wealth and power and were very tightly coupled with the government. It is WELL worth your while to consult Mike Duncan's Revolutions podcast, - the first five chapters of season 3 address this brilliantly.

So among the nation-states we have increasing complexity, increasing civilian power, political power divided between an executive system that is unable to confront the complexity of modern life, and a parliamentary/representative power that hasn't figured out how to limit and control power. The system is broken.

While there are potentially multiple different solutions to the problem, they would require a book to explore. Most of the solutions however involve a redistribution of effective power.

Or, returning to the one sentence answer, the growth in the complexity, power and agency of contemporary society exceeded the ability of an ancien regime monarchy to control; as a consequence a new form of governance was needed that would re-architect the exercise and coordination of power. Individuals of the middling sort had access to economic power that exceeded that exercised by nobility in prior generations.

In closing, I'll note that I really quite sincerely did try to be brief. But the questions are complex.