And there is an essential continuity among the national variants of the Enlightenment in their secularism.
By secularism, I don't mean atheism. Atheism is a philosophical position. I hold to it, and I regard the spread of atheism over the long term as both likely and overwhelmingly beneficial. But for cultures born of the Enlightenment, private religious belief is not so much an enemy as an irrelevance. What matters is that religion should not intrude into the public sphere. It must make its accommodation, however it chooses to get there, with modern mores, liberal values and secular education. If it won't, then it makes itself an enemy.
In fact, however, it is very difficult to show that this is the case: to a very large degree this is an anachronistic imposition. The major thinkers of the Scottish Enlightenment, for instance, split clearly into people who might be called secularists (Hume, Reid, Gibbon) and people who can in no serious sense of the term be called secularists, despite clearly being Enlightenment thinkers (Reid, Campbell, Witherspoon, Beattie). The French Revolution created what was literally a state religion of Reason. Enlightened Orthodoxy in Geneva was hardly secularist. The famous English 'Moderation' was largely furthered by people like Warburton, who was very far from thinking that private religious belief was an "irrelevance" or that it should not intrude into the public sphere: their Moderation was the middle way of the Church of England (between the two extremes of priest-dominated Catholics and visionary Quakers and Methodists). Actual advocates of separation of church and state, outside of certain religious groups like the Baptists, were relatively rare if all of Europe is taken into account; some on the secularist side, in fact, proposed establishmentarianism as the means for de-fanging religion (Hume is an example), and a major French Enlightenment document like the Civil Constitution of the Clergy is not at all something that banishes religion from the public sphere, despite secularizing it. There are no doubt particular thinkers of whom Kamm's summary would be a reasonable summary. But the 'Enlightenments' were hardly one-note music boxes, and on the topic of religion the notes make for extraordinarily diverse polyphony. (I think David Sorkin's The Religious Enlightenment does a fairly decent job of looking at some strands of the Enlightenment that are often overlooked.) And even the Virginia Statute for Religious Freedom, which Kamm holds up as a key example, doesn't lay down the public/private distinction the way Kamm does: "all men shall be free to profess, and by argument to maintain, their opinions in matters of Religion, and that the same shall in no wise diminish, enlarge or affect their civil capacities" is not a recipe for banishing religion from public view, and it protects public religious expression as much as it limits it. This is one of the possible effects of religious liberty and separation of church and state: no longer enforceable by government, religion still pervades the public sphere because impediments preventing it from participating in the arguments of the public sphere are removed, and therefore it becomes something that regularly has to be dealt with by politicians in their attempts to persuade their fellow citizens. You can take away the force of law from every religion; but if the mind of man is free, that includes being free to bring up religious claims in civic and political arguments. And Americans, to take just the obvious example, always have.
Kamm opens by proposing the Virginia Statute as the most significant document of the Enlightenment. My vote for the most (historically) significant document of the Enlightenment: Rousseau's Emile, the novel that launched a thousand educational reform movements.