Oh, thank you both very much, Bookbox and horselydown, and I do apologise for posting that extraneous work! It is me who was going batty, clearly.

By the time I got to looking for the last time, I was tired after hours of peering at the three pages and must have zoomed in at the wrong spot. I wondered why I hadn't spotted those names before!
So as far as I can see so far, the only named relatives in the will are a single son, John; daughter Alice married to Nicholas Chapman and their son John; daughter Margaret married to David Sterne and their son Giles; and sister Johane/Joan (interesting about the derivation of that one, btw horselydown - I never did Latin at school, but have always found languages and etymology fascinating!).
And then there are the friends Richard Amherst and Sir Thomas Springett. David Sterne is one of the witnesses, I think Arthur Langwart/Langward(?) another one.
I am now going to put together the translations I have, then review it all again and see if I spot anything else of potential interest. (A shame I don't have a wife's name or surname, because there are no obvious matches in the Sussex marriage index, but I will cross that bridge later...)