The law of large numbers is not a statement about probability in the intuitive sense, it's a statement about functions which satisfy the Kolmogorov axioms. Such functions don't necessarily have to have anything to do with frequencies or statistics.
For example, consider the interval $[0, 1]$. Define a function $L$ on measurable subsets of this interval by $L(A)=\text{length of $A$}$. Notice that I'm not imposing any kind of interpretation on $L$ as being a "probability", like the probability of hitting $A$ if you chucked a dart at the interval, or something like that. It's just the geometrical length, but nevertheless it's easy to see that it satisfies the Kolmogorov axioms. Now, under the function $L$, it turns out that $d_n(x) = \text{the $n$-th binary digit of $x$}$ defines a sequence of i.i.d. Bernoulli random variables, each with "probability" $0.5$, which in our case just means that the length of the set on which $d_n=1$ is $0.5$. The law of large numbers now tells us that the length of the set of numbers whose binary expansions have a equal asymptotic proportions of ones and zeroes is $1$. Notice how statistics never enter the picture: we're just doing geometry.
We can try to connect Kolmogorov's axioms with frequencies by saying something like this:
Consider some repeatable experiment. To say that a given set of outcomes has probability $p$ means that if you repeat the experiment a very large number of times, you'll get an outcome in that set approximately $p$ of the time.
How we can justify such a prediction is a separate matter. The point is: if we do assume this prediction, what can be deduced from it?
If we assume the above, then the function $P(A) = \text{long term frequency of $A$}$ satisfies the Kolmogorov axioms (kind of... if we table the discussion of issues like countable additivity and what "approximately" and "very large" mean).
As you point out, this seems like it's equivalent to just assuming the law of large numbers. But that's not quite the case. The LLN actually allows us to relax the above prediction slightly.
Let's say we want to apply the LLN to a coin flip. In that case, the "repeatable experiment" in question is the experiment of "tossing a coin a very large number of times", let's call this a "flip series". The LLN is then a statement about Kolmogorov functions on the set of all possible flip series outcomes.
Now of course, if we just assume that if we flip the coin a large number of times, we'll get heads about half the time, this is equivalent to assuming the LLN. But thanks to the LLN, we can assume something slightly weaker than that, and get LLN as a logical consequence. Namely, we only need the following assumption:
If I perform a large number of flip series, and only look at the outcomes of the $n$-th coin each time, then that coin will turn up heads about half the time.
Essentially, if you perform many many flip series, and represent the results in a table where each row is one series, like this:
$$
HHHHTTHTHTTTHTHHHTH... \\
TTHTHHTHHTHTHHTHTHT... \\
THHHTHTHTHHTHTHHHTT... \\
THTHTHHTHTHTHTHTHHT... \\
THTHTHTHTHHTHTHHTHT... \\
\vdots
$$
Then the assumption you're making is that there are approximately $50\%$ heads in each column, and the LLN (if you also make the assumption of independence) allows you to conclude that therefore, there are also approximately $50\%$ heads in each row.
Admittedly, if you're going to make an assumption about the $n$-th coin flip in the series, it seems like you may as well make the same assumption about the flip series themselves. But remember that just because random variables are i.i.d., doesn't mean they have to represent the same physical experiment. For example, imagine you had a big box of lots of different coins, and they were all numbered, so they were distinguishable. In that case, the LLN allows you to convert a set of assumptions about the individual coins into a conclusion about all of the coins in aggregate.