I don’t think France successfully became a democracy at all during the Revolution; the First Republic quickly fell into the mass murdering reign of terror under Robespierre, who was ousted by the Directory, who remained in power for four years until Napoleon’s coup. Lafayette himself was basically driven out of France by the radicals.
On the other hand, Britain itself was already fairly democratic by 1776, and the thirteen colonies had their own representative government institutions. The point of complaint was that the British were simply ignoring those institutions and flagrantly violating the traditional rights that Englishmen were entitled to. And a lot of the British were sympathetic to this viewpoint so there wasn’t the political capital to crack down on the rebellion as harshly as Britain would do in various other colonies at other points in time.
And aside from Lafayette, the French monarchy—the same absolute monarchy the Revolution sought to overthrow—was the regime that supported American independence, mostly out of “enemy of my enemy is my friend” considerations.