I didn't look too hard at this, but:
John "the Leather Seller" is reported to have baptised son Harmon 3 Oct. 1613 and John 24 Dec. 1614 at St. Martin's in London. FindMyPast shows transcriptions of these baptisms at Westminster, but the transcription does not indicate the father's name. The children's names are Harmon Atwood and John Atwood.
Alicia Crane Williams says only that Elijah Francis Atwood provided no evidence of the parental links that he claimed. He did provide some vital records, so what is missing is really the proper correlation that this and that records refering to "John Atwood" are referring to the same person.
ACW does note that it is possible John Atwood the Leather Seller was father to Harmon, and notes Pope's Pioneers as the reference. If this is credible, and if there is a credible case to be made that this Harmon was the same baptized at Westminster in 1614, then it seems clear that at least he had a brother John.
But there would be a lot to prove here:
* Harmon who appeared in Boston can be reasonably attributed the baptismal record at Westminster
* John of Plymouth was brother of Harmon. For this, Elijah Francis Atwood cites a 1631 lawsuit where John sued his brother Harmon. Some effort should be made to track this down.
* Even then, the connection of John of Plymouth as brother to Henry and Stephen is based tenuously on coincidence of land and freeman records for timing and proximity, as well as family lore. It is not an unreasonable deduction, but it makes it even more important to shore up the first two bullet points.
Y-DNA would not help determine if Henry, Stephen, John, and Harmon were brothers versus, say, cousins or some other sort of kin. But it would be very useful if it could show that Harmon was related to one of the others, since that seems to be the weakest link. Are there Harmon descendants who have tested the Y-chromosome?