He doesn't have a problem with Gates's statement, he has a problem with Isaacson's lack of research. Gates didn't write the book, Isaacson did. This is just one example where Jobs says something true, Isaacson thinks he's lying and instead trusts someone else who are either lying, or don't know the truth.
Who is Isaacson trusting? Bill Gates? If so why publish text that Gates disagrees with the truth of. Amelio? If so why publish the fact that Gates disputes that version? It sounds like he doesn't really "trust" either account, and why should he?
Gruber presents this as the best example of Issacson not trusting something true Jobs said and "trusting" the lies of others (as do you) but really it's just nerdy nitpicking about how far you should emphasise and editorialise the subjectivity of 3rd party accounts, there's no quote from Jobs being disproved in this example it's just Amelio (who in the anecdote just picked NeXT over NT!) describing Gates angry reaction at the time and Gates commenting about it later. Gruber's reaction is a total non-sequitur. He can't cope with an angry outburst by a rival that's just lost a business deal being left unchallenged, when the context is clear.