The Trinitarian case for Universalism takes a piece from Arminian logic on the atonement, a piece from Calvinism (indeed a reductio against universal atonement) and puts them together to teach that result. Arminianism teaches Christ made a universal atonement, as opposed to a limited one. Calvinism says Christ died for His Elect only, else His blood would be wasted on the unsaved. Arminianism teaches Christ died for the unsaved too; they just reject His grace. The Trinitarian Universalists seemed to argue that 1) Christ made a universal, not limited atonement. And 2) indeed, His blood WOULD be wasted if even one single soul for whom He died was not saved. Hence, everyone is saved eventually.
I found a source of more letters from Benjamin Rush explaining the case for Trinitarian Universalism, in particular letters of his to Universalist guru, Rev. Elhanan Winchester.
Your funeral sermon for Mr. John Wesley does honor to the philanthropy of your universal principles. I admire and honor that great man above any man that has lived since the time of the Apostles: his writings will ere long revive in support of our doctrine---for if Christ died for all, as Mr. Wesley always taught, it will soon appear a necessary consequence that all shall be saved.
-- To ELHANAN WINCHESTER, November 12, 1791.