Everything in the Oxfordshire visitation has checked out in surprising detail. I think the Yorks. pedigree is good back to this first guy, Sir William Anne, the Constable. The primary sources show that the Haringell marriage is a real thing, that Alexander Anne of Frickley was a real guy (fl. through 1370s), that Ralph Anne of Frickley was a real guy (fl. 1410, d 1454). Sir William was a real guy as well, but he was probably well dead by the time Alexander Anne was born, so I think we have a classic case of skipped generation. Sir William is probably the father of whoever Alice Haringell married. Who that was, I don't know. There are definitely a few Annes running around in the mid 1300s though, so there are some candidates out there. You know, maybe this elder son John of William's who is supposed to have died young actually belongs in the main line. It seems kind of odd to me that the pedigree preserves this account of him when it obviously ignores many subsequent generations of younger sons. I believe I saw some mention of a John de Aune in the mid 14th C in the patent rolls.
The one I'm really confused about is this alleged Sir John Anne of Frickley who supposedly died in 1441 and populates all these internet genealogies. I can't find a trace of anyone even vaguely resembling this guy, and if he really was a knight, that shouldn't happen. Yeah, if he was a guy who drove a dung cart for a living, he could easily have slipped past the notice of history, but knights get recorded. Maybe not in great detail, but there should be something.