That link does seem to be broken.
Yes they are confused.
There were 8 successive William Gascoignes of Gawthorpe, starting with the judge, almost all knighted.
This wasn't always clear. Early writers confused the generations. Somebody combined all 8 into one and said the judge left an heiress who married a Wentworth. Dates didn't used to be their strong point.
What is known about the family before the judge is hardly worth knowing. The judge's father or grandfather William leased Gawthorpe ca 1360s when Harewood fell out of the Lisle of Rougemont barony.
A William G of Kirkby Wharfe, probably the judge's grandfather, smashed up the premises of a business rival at Wainfleet. I guess they were importers of Bordeaux (then called Gascoyne wine) and Gascoyne was his trade name. Edward II encouraged the trade, being a big fan of all things Gascon.
A William G is named as a witness in a proof of age, but these things are pretty worthless.
The early part of the traditional pedigree is a fake, probably concocted by Richard Gascoigne the antiquarian. It extends the line of 8 Williams to 15, starting with the obligatory brave Norman knight (well at least he had a French name), and with the judge in the middle at number 8.
John Jones gives it in more or less original form. I don't suppose he believed it, but he had high-powered subscribers.
Agnes de Gawkthorpe is obviously put in to be the heiress of Gawthorpe. But no Gawkthorpes ever had the place, they came from somewhere else.
The Bolton and Frank wives have a sort of ring of truth, but perhaps that's what we're supposed to think.
Cracks appeared. There aren't enough generations to go back to the Conquest, so extra generations of Williams were added, up to 19 total. This changed all the numbers.
Other people dropped the early generations and started with the judge, or 1 or 2 or 3 generations earlier - 4 more different numbering schemes.
Some people reshuffled the wives. Somebody decided Agnes was no good and made her a Goldthorpe. Then some other trusting soul showed both.
Nobody had any evidence for their improvements. There's no truth to be had by trying to reconcile all this stuff.
On WikiTree, John Cherry has tidied the line up recently and restored the original 1-15 numbering. Let's hope it sticks.
The judge and his dates and wives and kids have been very vexed questions, especially Agnes Constable. But I haven't seen Richardson's conclusions.
Richard Gascoigne may also have had a hand in the fake Wentworth pedigree. When he died, his papers were quickly acquired by the Earl of Strafford and burnt, on legal advice. (Illegal advice?)
As for John Savile, his traditional pedigree is another fantasy that comes in 57 varieties, owing to everybody trying to fix it by guesswork.
Above 1325 it was debunked a century ago, but it's still all over the internet. After that, most people seem to have assumed it was OK, but actually the wives don't stand up. Isabel de Radcliffe, Isabel Fitzwilliam, Margaret Pilkington all look mythical, and there isn't much to be said for Alice Gascoigne except that she's still generally accepted.