The enigma of America's secular roots Joel Barlow's disavowal of Christianity as the basis for US government in the 1797 treaty of Tripoli is a mystery

Nonsense. While the exact reason for Joel Barlow’s choice of words is contestable, American secularism predates the Treaty of Tripoli. The 1789 Constitution states that there shall be no religious test for public office. In 1791, the Bill of Rights guaranteed Congress would make no law respecting an establishment of religion or prohibiting the free exercise thereof. Only in being secularists were the Founding Fathers in agreement on any religious question. For example, Jefferson was the first to talk of building “a wall” between Church and State. That controversy didn’t attend the Treaty of Tripoli’s wording is also proof secularism was already in force in the US.

it is clear that he had once had religion, and lost it.

No it isn’t. All this article goes on to tell us about Barlow’s comments regarding religion is he disapproved of organised religion and approved of secularism. This doesn’t mean he didn’t have religious beliefs or practices.

In contrast to the militant secularists of today, whose work suggests that ignorance and feeble individual minds lead to religion, Barlow thought that sick societies led to religion.

Who says ignorance & feeble individual minds lead to religion? If only “militant secularists” really existed, this straw man might be slander. Fortunately, another straw man protects it from prosecution.

moderate secularists today are quick to concede the formal beauty or theoretical appeal of religion.

What on Earth do either of these mean? Making claims unsupported by evidence is bad, end of.

Barlow thought that a religion or "mode of worship" granted "any preference in the eye of the law" was incompatible with "equal rights". Therein lay the force and the fury that drove the first generation of American secularists. Religion, they insisted, was responsible for inequality.

More accurately, there was inequality regarding how society treated people on the basis of their faith. To say religion itself caused the inequality is a separate claim, though many of the 18th century secularists may well have agreed to it. But as a political movement, the point of secularism was that it was wrong not to treat equally people of all faiths and none.

Barlow's outspoken radical views

Why is “let’s treat people fairly” radical, either in the sense of extremism or novelty? Why is it only those who don’t say what organised religion wants to here who are called outspoken? The things said by American Christians against non–Christians, gays, abortionists etc. while in or running for office are routinely horrendous.

in the United States, secularism and the fight for equality have parted ways

No they haven’t. Secularism is one of the requirements of equality. No doubt Haselby wants us to believe this is not so with modern secularists. But which specific inequalities advocated by modern secularists can he think to provide as examples? Well, none.