Why should the frequency of heads in a coin toss converge to anything at all? Suppose we have any kind of coin. Why should the relative frequency of getting a heads converge to any value at all?
One answer is that this is simply what we empirically observe this to be the case, and I think this is a valid answer.
However, my question is, is there some simple set of principles/axioms that apply to coin tossing from which we can derive this fact?

If you assume the coin tosses are independent of each other, and that you are equally likely to obtain heads on any one coin toss, then it isn't an axiom, and in fact follows from the strong law of large numbers.
EDIT: Just to answer some of the other things in your post, statistics is built upon probability theory, so if we empirically observe the frequency of heads converging to a specific value then that's fine, but we would like the empirical observations to agree with our probabilistic model.
And just to clear up a potential confusion before it arises, the strong law of large numbers isn't just related to coin tossing, and is an incredibly versatile and useful theorem (for example, under the same assumptions listed above, we could apply the same argument to dice throwing, and even continuous measurements!)
