You asked if General John Bates was descendant of immigrant John Bates (1598-1666)
Per DNA, the answer is no.
DNA KIT NO. 2060 Descendant of Julius Bates (1780-1864), who was brother of General John Bates, participated in the Bates DNA Project, and his DNA did not match DNA from donors who descend from immigrant John Bates (1598-1666)
KIT No. 6122 He did have an interesting DNA match.He matched a titled Englishman, now deceased.