I believe it is misguided to think that classical probability makes sense any more than quantum mechanics, with its "peculiar" probability calculations, makes sense.

I'm going to be slightly mischievous here and make a friendly attack your first paragraph: does really make sense?

Of course it makes perfect sense as a measure-theoretic definition, but how do you know it represents probabilities for real world random events? What does "probability" even mean? Do you take a frequentist, or subjectivist standpoint in bringing meaning to the word? I do think that we only "understand" classical probability insofar that we are simply accustomed to it.

The way wherein the first paragraph makes sense is due, I believe, mainly to Kolmogorov. His grand contribution was to understand that measure theory gives us a way to rigorously, set theoretically define events and show that the calculation of "probabilities" through their measure gives us a mathematical system that reproduces Pascal's, Laplace's and so forth intuitions about probability.

You can look at Kolmogorov as a physicist here: he's making postulates that events won't be represented by things such as Vitali sets, and that Pascal's intuitions, in keeping with the law of excluded middle that, are reasonable.

But along come the experimental physicists and show, experimentally, that this framework does not model all situations in the experimental physics world. In particular, the Bell Inequality, which is a Fréchet Inequality, can be violated. There are propositions that cannot be classically joined by the "and" operator: $X$ has momentum $p$ AND $X$ has position $x$ has no meaning in Nature. $\sigma$- and Boolean algebras simply cannot describe real world systems, and this is an experimental fact. See Valter Moretti's answer to the Physics SE question "Classical logic in concern with QM Mathematics".

Quantum observables make sense because they foretell experimentally measured results, whereas classical probability does not. The latter is experimentally falsified.

I believe it is misguided to think that classical probability makes sense any more than quantum mechanics, with its "peculiar" probability calculations, makes sense.

I'm going to be slightly mischievous here and make a friendly attack your first paragraph: does really make sense?

Of course it makes perfect sense as a measure-theoretic definition, but how do you know it represents probabilities for real world random events? What does "probability" even mean? Do you take a frequentist, or subjectivist standpoint in bringing meaning to the word? I do think that we only "understand" classical probability insofar that we are simply accustomed to it.

The way wherein the first paragraph makes sense is due, I believe, mainly to Kolmogorov. His grand contribution was to understand that measure theory gives us a way to rigorously, set theoretically define events and show that the calculation of "probabilities" through their measure gives us a mathematical system that reproduces Pascal's, Laplace's and so forth intuitions about probability.

You can look at Kolmogorov as a physicist here: he's making postulates that events won't be represented by things such as Vitali sets, and that Pascal's intuitions, in keeping with the law of excluded middle that, are reasonable.

But along come the experimental physicists and show, experimentally, that this framework does not model all situations in the experimental physics world. In particular, the Bell Inequality, which is a Fréchet Inequality, can be violated. There are propositions that cannot be classically joined by the "and" operator: $X$ has momentum $p$ AND $X$ has position $x$ has no meaning in Nature. $\sigma$- and Boolean algebras simply cannot describe real world systems, and this is an experimental fact. See Valter Moretti's answer to the Physics SE question "Classical logic in concern with QM Mathematics".

Quantum observables make sense because they foretell experimentally measured results, whereas classical probability does not. The latter is experimentally falsified.