There was some discussion about this on the FOM mailing list: see for example https://cs.nyu.edu/pipermail/fom/1999-April/002983.html, where someone claimed Wiles needed in accessible cardinals, and Harvey Friedman essentially told them to stop being stupid. (Friedman is the guy who created the subject of reverse mathematics, which studies what axioms are necessary for any given result.)
An analogy would be the claim that classifying groups of order 4 needs Grothendieck universes, because groups of order 4 form a proper class, so isomorphism classes of groups of order 4 form a 2-class of classes. This is obviously silly: it is trivial to restate the classification of groups of order 4 without using proper classes, but this makes the statement slightly more complicated: you have to talk about groups that are hereditarily finite sets or something like that, which is just an irrelevant complication. The use of Grothendieck universes is similar: for example, the collection of all etale spaces over a scheme is a proper class so in some sense uses universes to construct it, but is equivalent to a much smaller set so the use of universes is not necessary.
