The Congregationalists (but certainly not the Quakers) might have bequeathed your church a more liturgical service . . .
By golly you are on the right track. Our congregation is directly descended from the Congregational Church.
Unitarian from Congregational.
Coincidentally, the first Protestant Church I ever attended was a Congregational Church.
From a historical point of view, the Unitarians grew out of the Congregationalist tradition. The old New England Congregationalists began essentially as Calvinists, but by the late 18th/early 19th c. had mostly morphed into Unitarians, and many of them were preaching Universalism. The Congregationalists eventually split into a traditional branch and a Unitarian branch. Guess your church was in the latter group.
I had sort of inklings of this from years ago when I took a course on the New England Transcendentalist writers, who all grew out of that tradition. That's why it seemed plausible.