OK, I think I understand this all now. The simple answer is, no, Edward Fitz Randolph the immigrant is not a known descendant John of Gaunt, Edward III or have any other known connection to the kings of England.
The connection comes from: L.V.H. Randolph. Fitz Randolph Traditions. (New York, 1907), p. 120. It is complete garbage. This book made the Fitz Randolph line go:
Edward Fitz Randolph -- Emigrant,
son of Edward Fitz Randolph of Langton Hall d. 1635
son of Christopher Fitz Randolph
son of Randolph de Neville
5th son of Ralph (or Randolph) de Neville -- Duke of Westmoreland d. 1565.
No, the Fitz Randolphs are not descended from the Dukes of Westmorland and the spectacular ancestry that would go with it. They are thought to descend from the Fitz Randolph family of Spennithorne, co. York (no connection to the Dukes of Westmorland).
This family had been traced back quite a few generations to Ranulph Fitz Robert who married Mary le Bigod, daughter of the Magna Carta Surety, Roger Le Bigod. The line appears in early editions of Magna Carta Sureties and in Douglas Richardson’s 1st editions of Magna Carta Ancestry. However, there is an unproven or doubtful generation (John Fitz Randolph who m. Joan Conyers is not proven to have had a son John) and the line has been removed from Richardson’s most recent work Royal Ancestry. The line remains in WikiTree.
NEHGR vol. 97 (1943), 275-277 and 295-298
NEHGR vol. 99 (1945):335-336.
Weis. Magna Carta Sureties 5th edition (1999): line 164, for old line. There is a good discussion of both the evidence and of the problems with the line by the noted genealogist John Insley Coddington
Richardson. Magna Carta Ancestry 1st ed. (2005):342-343, for old line.
See Richardson. Royal Ancestry vol. IV (2013):226, sub Neville. showing the line removed.