I don't think you're quite right. "Carrier" is absolutely a mathematical term, referring to the set of elements of a group, as opposed to the group operation. That is, a group is defined as a binary operation satisfying the group axioms, on the elements of a certain carrier set.

For those unfamiliar with groups, one way to comprehend groups (by the Cayley theorem) is to essentially imagine groups to be sets of permutations (and in this case, the Carrier set would be the set of permutations, with the group operation being composition of permutations).

I'm not sure I'm any convinced by your argument of the collection of all groups being a group either, and whether that was what was referred to by the author. In any case, I don't think that follows the usual form of the Russell's paradox or Girard's paradox. I'm fairly certain that the "warning light" that the author mentions in relation to the set of all groups is about the set being too large to be consistently considered a set, rather than anything circularly related to groups.