Some of them came for those reasons. Other came for more secular purposes -- the original English settlement in North America was, of course, not Puritan Plymouth but rather Jamestown, Virginia, the profit-seeking venture of a joint-stock company.
And even those who did come seeking religious freedom tended to think about that concept in very different terms than we do today... Our histories tend to de-emphasize things like the fact that Puritan minister and Massachusetts leader Cotton Mather, for example, wanted to capture any Quakers who showed up in the colony and sell them into slavery in the West Indian sugarcane fields. That kind of "religious freedom" was narrowly and, indeed, violently sectarian. Which is one reason why many of the most fervent supporters of the separation of church and state by the time the US gained independence were deeply religious people who understood that Massachusetts-style intermingling of religious and state authority was *bad for religion.*
Today we almost always tend to think of "separation of church and state" as something designed to protect government from religion... but I think its more powerful effect over 200+ years of American history has actually been to protect religion from the worldly corruption of politics. I actually would argue that the reason why the modern United States is a such a remarkably religious country (compared to all other advanced Western democracies), the reason why even I, as an agnostic, think there is some validity to describing us as a "Judeo-Christian Nation" is precisely because we have been careful to defend the principle of secularism in the public sphere. I think the would-be theocrats of the so-called Religious Right (a recent phenomenon of the last 3 decades, really) better be careful they don't gain the world but lose their soul; we might end up like France, a country where everybody is nominally Catholic, a member of the state-sponsored church, but almost no one actually attends to mass and religion plays a minor role in shaping modern society.
Politics = corruption. Breaking down the wall between church and state is bad *for the church.*
SR