Thanks for your post and being cautious about your genealogy.My 4th great-grandfather was William Steed (c1743-1824) who was a brother of the Captain John Steed you mentioned. Both were sons of Peter Steed (c1720-c1780).
Captain John Steed, born in New Jersey 1746, died June 1800 in Hampshire County, VA with no issue. Under Virginia law of primogeniture his properties were inherited by my direct-line ancestor, William Steed, who was the eldest living brother at the time. The younger full-blood brother, James Steed (c1755-1828) received no property - Jean Glass, RIP, thought that James Steed was born in 1740 since his wife, Ann, was born in that year, but James received nothing. A half-brother, Aron Steed, received no property under the law either.
John Steed is the captain in the Revolutionary War (8th Virginia and later, 4th Regiment) that many people confuse with John Steed of Brunswick Co, VA (c 1715-1789). Descendents of the older John Steed (who never served in the Revolution) hijacked my ancestors' records to apply for DAR membership which was since disallowed. They were eventually able to regain DAR membership since the older John Steed made claim for some property that was confiscated by troops during the war. The confusion that was started keeps floating around the internet.
Bad news: I'm not aware of any documents that mention who Jesse Steed's parents were. However, there is a Thomas Steed who was born in New Jersey in 1759 who wound up entering the American Revolution in Bedford County, VA at the beginning of the war, and may be related to Jesse. We don't know yet who his parents were either.
Good news: I suspect that the Pennsylvania and Northern Virginia Steeds passed through New Jersey during the mid 1700s and originated in New York and before that, possibly Connecticut. I have reasons for believing this, but haven't been able to nail down sufficient documentary proof yet.
In 1761 Peter Steed (my line) and Henry Steed (brother?) were in Northampton County, PA on the Tax List for Lehigh and Macungie townships respectively. Peter moved his family to the Maryland-Pennsylvania border by 1765 and his son John Steed, who had entered the French & Indian War in 1763 in Northampton County, PA wound up settling in the same area.An educated guess is that Jesse Steed was a nephew of Peter Steed (c1720-c1780), but we don't know yet who his father was.
I am looking for brothers of Peter Steed (aside from Henry mentioned above). There was a George Steed living in Elizabeth Town, NJ (Essex County at the time) in 1759, and a John Steed who married Katherine Cox in Philadelphia in 1742. Either one could be Jesse's parent(s).
Ivan Steed who wrote the 1957 Steed book never figured it out, Jean Glass never figured out Jesse either. Maybe with additional documents coming on line every year, we will have been luck.