I. I agree with you that no version of the Law of Large Numbers tells us something about real life frequencies, already for the reason that no purely mathematical statement tells us anything about real life at all, without first giving the mathematical objects in it a "real life interpretation" (which never can be stated, let alone "proven", within mathematics itself).
Rather, I think of the LLN as something which, within any useful mathematical model of probabilities and statistical experiments, should hold true! In the sense that: If you show me a new set of axioms for probability theory, which you claim have some use as a model for real life dice rolling etc.; and those axioms do not imply some version of the Law of Large Numbers -- then I would dismiss your axiom system, and I think so should you.
II. Most people would agree there is a real life experiment which we can call "tossing a fair coin" (or "rolling a fair die", "spinning a fair roulette wheel" ...), where we have a clearly defined finite set of outcomes, none of the outcomes is more likely than any other, we can repeat the experiment as many times as we want, and the outcome of the next experiment has nothing to do with any outcome we have so far.
And we could be interested in questions like: Should I play this game where I win/lose this much money in case ... happens? Is it more likely that after a hundred rolls, the added number on the dice is between 370 and 380, or between 345 and 350? Etc.
To gather quantitative insight into answering these questions, we need to model the real life experiment with a mathematical theory. One can debate (but again, such a debate happens outside of mathematics) what such a model could tell us, whether it could tell us something with certainty, whatever that might mean; but most people would agree that it seems we can get some insight here by doing some kind of math.
Indeed, we are looking for two things which only together have any chance to be of use for real life: namely, a "purely" mathematical theory, together with a real life interpretation (like a translation table) thereof, which allows us to perform the routine we (should) always do:
Step 1: Translate our real life question into a question in the mathematical model.
Step 2: Use our math skills to answer the question within the model.
Step 3: Translate that answer back into the real life interpretation.
The axioms of probability, as for example Kolmogorov's, do that: They provide us with a mathematical model which will give out very concrete answers. As with every mathematical model, those concrete answers -- say, $P(\bar X_{100} \in [3.45,3.5]) > P(\bar X_{100} \in [3.7,3.8])$ -- are absolutely true within the mathematical theory (foundational issues a la Gödel aside for now). They also come with a standard interpretation (or maybe, a standard set of interpretations, one for each philosophical school). None of these interpretations are justifiable by mathematics itself; and what any result of the theory (like $P(\bar X_{100} \in [3.45,3.5]) > P(\bar X_{100} \in [3.7,3.8])$) tells us about our real life experiment is not a mathematical question. It is philosophical, and very much up to debate. Maybe a frequentist would say, this means that if you roll 100 dice again and again (i.e. performing kind of a meta-experiment, where each individual experiment is already 100 "atomic experiments" averaged), then the relative frequency of ... is greater than the relative frequency of ... . Maybe a Bayesian would say, well it means that if you have some money to spare, and somebody gives you the alternative to bet on this or that outcome, you should bet on this, and not that. Etc.
III. Now consider the following statement, which I claim would be accepted by almost everyone:
( $\ast$ ) "If you repeat a real life experiment of the above kind many times, then the sample means should converge to (become a better and better approximation of) the ideal mean".
A frequentist might smirkingly accept ($\ast$), but quip that it's is true by definition, because he might claim that any definition of such an "ideal mean" beyond "what the sample means converge to" is meaningless. A Bayesian might explain the "ideal mean" as, well you know, the average -- like if you put it in a histogram, see, here is the centre of weight -- the outcome you would bet on -- you know! And she might be content with that. And she would say, yes, of course that is related to relative frequencies exactly in the sense of ($\ast$).
I want to strees that ($\ast$) is not a mathematical statement. It is a statement about real life experiments, which we claim to be true, although we might not agree on why we do so: depending on your philosophical background, you can see it as a tautology or not, but even if you do it is not a mathematical tautology (it's not a mathematical statement at all), just maybe a philosophical one.
And now let's say we do want a model-plus-translation-table for our experiments from paragraph II. Such a model should contain an object which models [i.e. whose "real life translation" is] one "atomic" experiment: that is the random variable $X$, or to be precise, an infinite collection of i.i.d. random variables $X_1, X_2, ...$.
It contains something which models "the actual sample mean after $100,1000, ..., n$ trials": that is $\bar X_n := \frac{1}{n}\sum_1^n X_i$.
And it contains something which models "an ideal mean": that is $\mu=EX$.
So with that model-plus-translation, we can now formulate, within such model, a statement (or set of related statements) which, under the standard translation, appear to say something akin to ($\ast$).
And that is the (or are the various forms of the) Law of Large Numbers. And they are true within the model, and they can be derived from the axioms of that model.
So I would say: The fact that they hold true e.g. in Kolmogorov's Axioms means that these axioms pass one of the most basic tests they should pass: We have a philosophical statement about the real world, ($\ast$), which we believe to be true, and of the various ways we can translate it into the mathematical model, those translations are true in the model. The LLN is not a surprising statement on a meta-mathematical level for the following reason: Any kind of model for probability which, when used as model for the above real life experiment, would not give out a result which is the mathematical analogy of statement ($\ast$), should be thrown out!
In other words: Of course good probability axioms give out the Law of Large Numbers. They are made so that they give them out. If somebody proposed a set of mathematical axioms, and a real-life-translation-guideline for the objects in there, and any model-internal version of ($\ast$) would be wrong -- then that model should be deemed useless (both by frequentists and Bayesians, just for different reasons) to model the above real life experiments.
IV. I want to finish by pointing out one instance where your argument seems contradictory, which, when exposed, might make what I write above more plausible to you.
Let me simplify an argument of yours like this:
(A) A mathematical statement like the LLN in itself can never make any statement about real life frequencies.
(B) Many sources claim that LLN does make statements about real life frequencies. So they must be implicitly assuming more.
(C) As an example, you exhibit a Kolmogorov quote about applying probability theory to the real world, and say that it "seems equivalent to introducing the weak law of large numbers in a particular, slightly different form, as an additional axiom."
I agree with (A) and (B). But (C) is where I want you to pause and think: Were we not in agreement, cf. (A), that no mathematical statement can ever tell us something about real life frequencies? Then what kind of "additional axiom" would say that? Whatever the otherwise mistaken sources in (B) are implicitly assuming, and Kolmogorov himself talks about in (C), it cannot just be an "additional axiom", at least not a mathematical one: Because one can throw in as many mathematical axioms as one wants, they will never bridge the fundamental gap in (A).
I claim the thing that all the sources in (B) are implicitly assuming, and what Kolmogorov talks about in (C), is not an additional axiom within the mathematical theory. It is the meta-mathematical translation / interpretation that I talk about above, which in itself is not mathematical, and in particular cannot be introduced as an additional axiom within the theory.
I claim, indeed, most sources are very careless, in that they totally forget the translation / interpretation part between real life and mathematical model, i.e. the bridge we need to cross the gap in (A); i.e. steps 1 and 3 in the routine explained in paragraph II. Of course it is taught in any beginner's class that any model in itself (i.e. without a translation, without steps 1 and 3) is useless, but it is commonly forgotten already in the non-statistical sciences, and more so in statistics, which leads to all kind of confusions. We spend so much time and effort on step 2 that we often forget steps 1 and 3; also, step 2 can be taught and learned and put on exams, but steps 1 and 3 not so well: they go beyond mathematics, seem to fit better into a science or philosophy class (although I doubt they get a good enough treatment there either). However, if we forget them, we are left with a bunch of axioms linking together almost meaningless symbols; and the remnants of meaning which we, as humans, cannot help applying to these symbols, quickly seem to be nothing but circular arguments.
Best Answer
You seem to be tackling several issues at once. First though, some inaccuracies. You write "when creating a system of axioms like these..." I'm not sure what 'these' refers to. Then you say "it's necessary the list of axioms is complete." Do you mean by 'complete' that there is only one model of the axioms (up to isomorphism)? if so, why is that necessary for modelling probability events? You comparison with the axioms of geometry is unclear as well. If you omit the fifth, you do not automatically get hyperbolic geometry, you can also get projective geometry. To claim that any of those is not what we wanted to have is peculiar, particularly from a modern perspective. Geometry encompasses much more than just Euclidean geometry. And again, even with the fifth there is not just one (up to isomorphism) Euclidean geometry, but infinitely many (of various dimensions).
Now I will try to address the question of what is so great about Kolmogorov's axiomatisation. The mathematics of probability is fraught with difficulties, both conceptual and technical. There are endless examples of seemingly simple questions that turn out to be very complicated or have severely counter intuitive answers (The Monty Hall paradox for instance). Problems that appear identical may turn out to be significantly different just because of changes in the protocol. In short, it's not easy.
Having said that, the probability theory of finite probability spaces is quite simple, at least in the sense that it is clear how to model finite probability spaces: Given a finite set of events, the probability of a subset of events is the ratio of that subset to the entire set. Sweet. From it flows quite a lot, but only when the total set of events is finite.
Often, the set of events is infinite. For instance, modelling throwing a dart at a dartboard is often done by imagining the dart board as a disk in $\mathbb R^2$, and then a throw of a dart corresponds to a choice of a point in the disk. Of course the disk has infinitely many points. What is the probability that the dart hits a given point, say the centre of the disk? Well, assuming the dart lands randomly at a uniform distribution over all points, the only possible answer is $0$. A point is just too small. This is already counter intuitive enough and raises the question of how to model all of this. Well, this is all related to the notion of how big a set is. An innocent question with a highly complicated answer. It's not simple at all to develop the theory that answers this question - measure theory. Issues related to the axiom of choice quickly creep up. A famous theorem of Vitali shows that it is impossible (assuming the axiom of choice) to meaningfully assign a measure to each and every subset of $\mathbb R$.
Now, measure theory was not developed to provide some foundations of probability theory. Instead it arose from questions of integrability. Kolmogorov's wonderful insight was that he realised the same formalism can be used to turn the intuition of what probability theory should be (as you say, pretty obvious axioms) into actual axioms. Before measure theory and Kolmogorov's seminal contribution nobody knew how to meaningfully and accurately work with infinite probability spaces. Thanks to Kolmogorov a formalism was born. Now that is truly wonderful.
Lastly, the paragraph you quote is talking about something all together different. Quantum mechanical considerations defy many conceptually obvious properties. Among them Kolmogorov's axiomatisation of probability. In the world of quantum mechanics even probability behaves differently than what we are used to. Such is life.