In some elementary analysis courses, we discussed what would fail without countable additivity, although it's not as if there would be some contradiction. It would merely be "not nice." We'd lose continuity under sequential monotone limits. Dunford and Schwartz say some things about finitely additive set functions. What happens when instead of notions like "$\sigma$ algebra" and countable additivity, you replace every occurrence of countability with some larger cardinal? So the reals may not be quite the right place to carry out such a theory because if you have an arbitrary sized disjoint union of "measurable" sets in the proposed theory, then unless all but countable many have "measure 0" we would have that the measure would have to be infinity or undefined in the signed case all the time. But what happens if more general values of the measures are taken, like say in a Banach space?
To the extent that the following question can be answered beyond "just because it worked out that way", why is it that our modern mathematical theories basically only care about scalar measures that are countably additive, or perhaps in strange situations banach space valued measures, but still countably additive? Why not finite, or why not bigger?