The major issue here appears to be that researchers are confusing two different families with the name of Pitt.
John Pitt of Blandford, Dorset was indeed the son of William Pitt and Helene (or Eleanor) de Havilland, William Pitt was, in turn, the son of Nicholas Pitt and his wife of uncertain name was the daughter of Roger Cooke. (Her first name may have been Elizabeth, but evidence is scarce).
This Pitt family was generally known as the Pitts of Blandford and Strathfieldsaye (to distinguish it from the Pitts of Curewiard, Perry near Stoke etc - and which is where the confusion arises). The descendants of the Blandford Pitts are firmly based in southern England - Dorset, Hampshire, Somersetshire etc.
There are very good sources attesting to this lineage - both primary and secondary. I would recommend the following:
* Burke's A General and Heraldic Dictionary of the Peerage and Baronetage, vol 2, p. 353 (via Google Books)
* Collins's Peerage of England; genealogical, biographical, and historical, Volume 7 (via Google Books)
* Pedigrees from the Visitation of Hampshire, pp 194-95, https://archive.org/details/pedigreesfromvis64beno/page/194
* The Haviland Genealogy, https://archive.org/details/havilandgenealog00fros/page/n21
* Will of Sir William Pitt d. 1636 (transcribed on http://gen.julianlyon.com/1636-sir-william-pitt/ and summarised on
* The Diary of William Hedges, The Diary of William Hedges, pp xxix and pp xxiix, https://ia800203.us.archive.org/7/items/diaryofwilliamhe783hedg/diaryofwilliamhe783hedg.pdf
* The Haviland Genealogy: 1653-1688, https://archive.org/stream/havilandgenealog00fros/havilandgenealog00fros_djvu.txt
* Will of William Pitt, merchant of Bristol (d. 1622), summarised on enster's Southern Genealogy, http://gensouth.blogspot.com/search/label/Pitt%20Family and transcribed on https://www.wikitree.com/wiki/Space:Will_of_William_Pitt_Merchant_of_Bristol_Feb.1524/5 ( a useful source for establishing the family name of Nicholas Pitt's wife, i.e. daughter of Robert Cooke)
The problem with the genealogical details on https://www.genealogy.com/forum/surnames/topics/pitts/1587/ is that its sources are unclear and it cites the Visitation of Shropshire, etc which do not support the data provided. I would not go so far as to say that it is "fictitious" but it is certainly confused. (I make it a rule never to consult genealogies on blogs or genealogies that fail to disclose sources, and I always double check sources before proceeding.)
Most sources agree that Nicholas Pitt is the first known ancestor of the Pitts of Blandford and Strathfielsaye and unless new information comes to light, it is doubtful that any researcher can push the lineage back past this point..
I hope that his helps.