No, I don't think it is true that both parties necessarily wind up with more accurate estimates after updating and agreeing, or even an estimate closer to what they would have obtained by sharing all their data.
That greatly diminishes the value of the theorem, and implies that it fails to justify blaming dishonesty and irrationality for the prevalance of persistent disagreements.
I'm not sure. Aumann's paper seems to only bill itself as leading to agreement - with relatively little discussion of the properties of what is eventually agreed upon. Anyway, I think you may be expecting too much from it; and I don't think it fails in the way that you say.
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