It makes sense, but it is no obligation.

The statement of the theorem is "whatever our function does, if it is consistent with the axioms, it is addition". This is used in the context of finite and quite imaginable amount of atoms. We could ascribe all of them equal valuation, but we can have some knowledge why some are more probable than other ones. But the proof requires us to have a lot of atoms and to be able to find as many equally-valued atoms as we need. Proving some inequalities with existing amount of atoms may need more atoms than we initially considered. Also, it may be that we know enough to give every atom a distinct valuation, in which case the proof just stops being applicable.

I have a counterexample even if we grant the existence of arbitrarily many atoms with the same valuation (by the way, it means that sum exceeds 1); I will describe it in the answer to another comment - I hope it is correct.