We argue using simple models that all successful practical uses of probabilities originate in quantum fluctuations in the microscopic physical world around us, often propagated to macroscopic scales

Their argument is that not only is quantum mechanics ontologically probabilistic, but that only ontologically probabilistic things can be successfully described by probabilities. This is obviously false (not to mention that nothing has actually been shown to be ontologically probabilistic in the first place).

Thus we claim there is no physically verified fully classical theory of probability.

They think they can get away with this claim because it can't even be tested in a quantum world. But you can still make classical simulations and see if probability works as it should, and it's obvious that it does. Their only argument is that it's simpler for probability to be entirely quantum, but they fail to consider situations where quantum effects do not actually affect the system (which we can simulate and test).