Why "countability" in definition of Lebesgue measures?

If $\sum_{I \in X} l(I)=\infty$, then there is a countable subset of $X$ that sum up to infinity too.

If $\sum_{I \in X} l(I)<\infty$, then only countably many terms of $l(I)$ is non-zero.

In either case, it'd just be the same as allowing only countable infinite terms.

It will work. The thing is, if $X$ is uncountable the sum is necessarily infinite. Thus only countable families $X$ will give approximations to the infimum, if it is indeed finite.