Issues with von Mises axioms of probability

Premise. More a long comment than an answer, but I felt compelled to post it since in my opinion this question deserve at least a possibly bad answer (I am not an expert in statistics nor in probability theory). Therefore I apologize from now if my language (or better the concepts I'll expose) will be somewhat hazy and mathematically imprecise.
Edit. Videtur the comments to this answer, I tried to improve it following temo's feedback: I hope to have succeeded in producing something at least minimally useful.

The three basic rigorous approaches. Historically, three different rigorous (in the sense of axiomatic) approaches to the theory of probability have been proposed:

1. The measure theoretical approach, by Andrei Kolmogorov. In this approach, probability is not defined in a direct way, but as a class of finite measures satisfying a handful of axioms. Thus it provides a means to identify probability distributions, not a direct path for their construction: if you get a set function in some unspecified manner, then you can check if it is a probability distribution or not.
2. The operational subjectivist approach, by Bruno De Finetti. This approach is constructive in the sense that its axioms aim to describe how to construct a probability. Precisely, in this theory, the probability is defined as the value a non biased and informed person could assign to whether a specific outcome is likely to occur. De Finetti proves the equivalence of his axiomatics to the classical ("Kolmogorov") probability theory in [1], which is unfortunately written in Italian: however, [2] is a nice technical review of a later work. In particular, a characteristic of the approach of De Finetti is the use of finitely additive measures, as described in [2].
3. The Frequentist approach was pursued by many scholars and Richard Von Mises was among them. The frequentist probability theorists define probability by means a limiting process on random samples which is reminiscent of the central limit theorem: Von Mises approach is based on the definition of some random sequences called kollektivs, according to [3], chapter 2.

Reference [3], especially chapter 2, is particularly pertinent to the our question since it describes why Von Mises approach has not been extensively pursued: the criticism of Paul Levy exposed at a conference on Probability theory held in Geneva in 1937, and his praise of Kolmogorov's approach, may have discouraged other scholars. On the other hand [3] also tries tho analyze Von Mises contribution in a deeper and less emotive way, so perhaps this is the right source to start with for an analysis of modern ramification of Von Mises's probability axiomatics.

Reference

[1] Bruno De Finetti, "Sul significato soggettivo della probabilità (On the subjective meaning of probability) ", (Italian), Fundamamenta Mathematicae 17, 298-329 (1931), JFM 57.0608.07, Zbl 0003.16303.

[2] D. A. Gillies, "Review: The Subjective Theory of Probability", The British Journal for the Philosophy of Science, Vol. 23, No. 2 (May, 1972), pp. 138-157.

[3] Michiel van Lambalgen, Random Sequences, Historical Dissertations HDS-08, Originally published: September 1987 (Amsterdam).