Recently I read a paper by Benci et al. describing an alternative to Kolmogorov's construction of probability where the probability measure $P$ takes values in a non-Archimedian field and we have $P(A) = 1 \iff A = \Omega$. One consequence of this is that now the definition $$P(A|B) := \frac{P(A \cap B)}{P(B)}$$ is valid for any $A, B \in 2^{\Omega}$, so we can avoid all of the rigamarole with conditional expectations that arises in the standard formulation, just like how in non-standard analysis we have that derivatives are actually quotients. We also can measure every element of $2^\Omega$ so we don't even need to bother with $\sigma$-algebras.
What I like so much about this is that it removes many of the initial hurdles to dealing with measure-theoretic framework, such how if $X \sim \mathcal N(0,1)$ then $P(X = 0)=P(X = \textrm{"blue"}) = 0$, i.e. we can't distinguish impossible outcomes from "almost never" outcomes.
My question: are there ever practical uses of non-Kolmogrovian probability, and if not, what's so special about Kolmogrov's framework that it's the only one that we can or do use to compute actual quantities of interest? By practical uses I mean real-life probability computations, like coin tosses and die rolls or like machine learning predictions such as estimating the conditional probability of defaulting on a loan.
My guess would be that it comes down to how all of these models agree on the finite cases (like the probability of getting 3 heads in a row for a coin toss) and it's only on the infinite and not-so-practical cases that we see disagreement (like the probability of tossing a coin and getting heads forever, and cases where $\sigma$-additivity can lead to counterintuitive results), but I'm not sure about this.
Update: @pash has pointed out that with hyperreal-valued extensions we have the transfer principle. Does this completely settle the issue? The Kolmogorov formulation has different axioms though so can't I expect that there are true statements in one formulation that are false in the other? And regardless, this still doesn't seem to answer my question of what is so special about Kolmogorov's formulation that it is the root of all of these non-Archimedian extensions.