In regard to Nathaniel Thayer and why there is so much confusion: Aside from the family geneaologist for the Thayer Family Association, Patricia Thayer Muno, who has cited her source for changing the parents of Nathaniel Thayer, from Richard, to William, there is also a book written on the Thayer family from Thornsbury, that, upon careful reading, you will find both of the Nathaniel Thayers, who were cousins, along with who the parents were of each Nathaniel.
In one of Patricia's books on the Thayer family, she had first documented Nathaniel, who married Abigail Harvey, with parent, Richard Thayer. Three years after she published her book, she realized there were two Nathaniels, and that the Nathaniel who married Abigail Harvey was actually the son of William Thayer and Mary Kellaway. Unfortunately, many family trees still contain the misinformation--showing Nathaniel as the son of Richard. Find A Grave was also infected with these errors, due to people either not following the update, or never getting it.
The book where you can find the Thayer information is: The Thayer Family of Thornbury: a study trying its reconstitution. Originally written in 1907, by Thomas Thayer Ojeda.
These two sources are better than anything that has come before, and are your best bet for correcting the profiles.