Sorry, it appears that we just are a long way from communicating clearly. I'll try again, a little:

What I outlined are the axioms of probability as started by
Kolmogorov in 1933 and based on Lebesgue's measure theory (of near 1900). So, Kolmogorov finds something in Lebesgue's theory, that is, a particular measure space that has everything we want for probability theory. In this way Kolmogorov shows that we can regard probability theory as just another measure space in measure theory.

Why do that? To get a different probability theory? Not really: Kolmogorov's start just gives essentially the same probability theory we had in 1932. It is just that in 1932, we had to talk about trials, events, probabilities, and random variables without being able to say, mathematically, what the heck they were. To see the importance here, back to near 1900 with B. Russell, etc. there was an effort to redefine everything in mathematics
starting with just sets. Then everything was constructed starting with just the low level ideas of sets. The serious result was axiomatic set theory, e.g., as in P. Suppes, Axiomatic Set Theory. So, numbers, functions, calculus, lines, planes, spheres, groups, rings, fields, vector spaces. etc. were all defined based on sets. And similarly for Lebesgue's measure theory. Then, after Kolmogorov, probability theory was also
defined based just on sets. Whew!

For what you want to do with probability theory in statistics, etc. as you mentioned, Kolmogorov's axioms should be of zero concern to you except you might feel
a little better knowing that the probability theory
you have been using all along does have a solid
foundation just on sets. So, you can go right along
with applied probability as you have been doing, and
Kolmogorov's axioms will essentially never get involved,
never help you and never hurt you.

For more, via the axioms, we can define trials, events,
probabilities, random variables, conditional probability,
stochastic processes, Markov processes, Gaussian random variables, Chi squared random variables, sufficient statistics, distributions, random vectors,
and on and on that you have already been knowing, loving,
and using.

But, maybe I spoke too soon: For Markov processes,
martingales, sufficient statistics, we very much
want the Radon-Nikodym theorem of measure theory
and use it for random variables, and we do. So,
without the Kolmogorov's axioms, we would be somewhat
stuck-o for Markov processes, etc.

> I think you'll find that it's hard to do even bread-and-butter statistical tasks like deciding how many subjects to include in a randomized trial without moving beyond axiomatic probability.

Not at all. There is no "moving beyond". Instead we
proceed essentially as we might have in 1932. That is,
in such work we rarely or never think about measurable spaces, measure spaces, sigma algebras, measurability, random variables as functions on the set of trials, etc. Again, we just continue to do applied probability and
statistics essentially as we might have in 1932.

Or, the Komogorov axioms are down in the sub, sub basement, and we rarely go there. For the rest of the structure, it is just the same or nearly so.

Okay?