Not necessary, because even if Leibniz did tell someone (such as his family) about calculus before publication, that doesn’t suggest that Leibniz/Newton did not discover calculus independently. Although the author does need to assume that Newton did not learn of calculus through someone that Leibniz might have told, that doesn’t require Leibniz not to have told anyone about calculus.

Although the author does need to assume that neither Newton nor Leibniz learned the details of calculus from a third person, that doesn’t require the author to think no third person ever discovered calculus. As long as that person did not share the details with Newton and Leibniz, the author’s reasoning still works.

Not necessary, because Newton’s belief about what Leibniz might have learned are irrelevant. We already know that the letter “did not reveal anything important about calculus.” Newton’s belief about what Leibniz learned doesn’t change the fact that the letter didn’t reveal anything important.

Not necessary, because even if Newton or Leibniz did know that the other had developed a version of calculus before Leibniz’s publication, what matters is whether one of them learned the details of calculus from the other or from a third person. You can know that someone else has discovered calculus, but still develop calculus independently as long as you don’t learn the details of calculus first.

Necessary, because if Newton or Leibniz DID learn crucial details of calculus from some third source, that raises the possibility that one of them did NOT independently discover calculus, which undermines the author’s argument. The author must assume that this didn’t happen in order to reach the conclusion that Newton and Leibniz each independently discovered calculus.