I should note that I suspect that's a general failure mode: take any utility function, and add an agent to the pool who has that utility function. That agent is now a candidate for the social welfare function, as it now satisfies the first two axioms and might satisfy the third. (Alternatively, appoint any agent already in the pool as the social welfare function; the first two axioms will be satisfied, and the third will be unchanged.)

That is correct. But in a case like that, the aggregate utility function is a linear combination of the original utility functions where all but one of the coefficients are 0. Being a linear combination of utility functions is not a strong enough requirement to rule out all bad aggregations.