Armitage seems to be an Anabaptist-influence historian, not a to-the-Jordan-successionist.
"The founders of Plymouth were not Puritans, or Non-conformists, but Separatists, who had paid a great price for their freedom, and had come from an independent congregation in Leyden."
"ROGER WILLIAMS, having adopted the old Baptist principle of absolute soul-liberty and given it practical effect in the civil provisions which he had devised, could not stop there. This deep moral truth carried with it certain logical outworkings concerning human duty as well as its rights, and as his doctrine could not stand alone in his thought, he was compelled to take another step forward. Relieved from all outside authority in matters of conscience, to which he had formerly submitted, he was now directly responsible to God for the correctness of his faith and practice, and by all that he had suffered he was bound to walk in an enlightened conscience. This compelled him to inquire what obedience God demanded of him personally, and threw him directly back upon his word as to his personal duty in the matter of baptism. While an infant he had been christened, but having now put himself under the supreme Headship of Christ, without the intervention of human authority, he found himself at a step on pure Baptist ground, and determined to be baptized on his own faith."
No succesionist that I know of would accept Williams as a true Baptist.