For instance, if I am trying to count my goats, the concept of natural integer allows me to count two groups, sum the results, and still have an accurate understanding of the number of goats I have.
As it turns out, the theory of natural integers allows one to define arbitrarily large numbers, even though I cannot possibly have more than, say, 1000 goats. Yet, a theory that deals only with numbers less than 1000 might be more cumbersome to work with. Try to define addition there and see for yourself.
The theory of natural numbers is useful because it provides a simple formal way to accurately count the number of goats (and possibly other things). It doesn't matter that it allows us to conceive of an absurd number of goats, like 1,000,000. It does the job, and as long you use it in the real world on real data you won't get absurd results.
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The same goes for the axiom of choice. Whether or not you accept it isn't going to make a practical difference once it comes to measuring the volume of helium needed to fill your balloon.
Think of it as upcasting and downcasting. You're dealing with a finite problem which is hard to solve. So you upcast it to a more general theory, solve it, and then downcast it back to the original setting.
There are as many ways to upcast as there are ways to pick independent axioms, so pick whichever make the problem easiest to solve, and don't worry about the "truth" of these propositions.