I can now say with complete confidence that this mystery is solved.
Obediah Cooper Spencer was the product of a "non-paternity event".
After forty years of genealogical, historical, and genetic research, including two DNA matches, it can now be shown that he was the biological son of a Mr. Hare (a son or grandson of Stephen Hare (1747-1845) of Haldimand, Northumberland, Ontario) by Ruth W. Cooper Spencer Weeks, and that he received the surname Spencer from his first step-father, who I believe to have been Charles Spencer of Prescott, Grenville, Ontario.
So the missing Mr. Spencer I have been seeking all these years was not from the Bay of Quinte after all.