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.
I've recently been getting into all of this wonderful Information Theory stuff and have come across a paper (thanks to John Salvatier) that was written by Kevin H. Knuth:
Foundations of Inference
The paper sets up some intuitive minimal axioms for quantifying power sets and then (seems to) use them to derive Bayesian probability theory, information gain, and Shannon Entropy. The paper also claims to use less assumptions than both Cox and Kolmogorov when choosing axioms. This seems like a significant foundation/unification. I'd like to hear whether others agree and what parts of the paper you think are the significant contributions.
If a 14 page paper is too long for you, I recommend skipping to the conclusion (starting at the bottom of page 12) where there is a nice picture representation of the axioms and a quick summary of what they imply.