I don't mean why is it important. I mean why can't we just define the "selector function" like $S\colon \mathbb{F} \to A, $ such that $S(X) = x \in X$, without an axiom?
Why can't we do that but we can, for example, take some set that satisfies a condition from an uncountable family?
As I understand your proposal, the problem is that it requires you to have a condition to construct that set.
In many situation it's certainly possible to do so and in those situation you need not to resort to the axiom of choice to guarantee the existence of a selector function. A vast amount of mathematics can get by without the need of the axiom of choice.
However if you have a situation when you can't guarantee that existence by a concrete example you will have a bit more trouble. In some special cases you could perhaps prove the existence without the use of AoC, but in general you would need to use the AoC to prove that.
There is proof that the AoC can't be proven from the other axioms.