The fundamental conceptual question about probability is …

To be provocative I often say that I have a very low opinion of the philosophy of probability literature. Here I do not mean technical specialist academic philosophy, but what I see in more popular discussions. This is not the place for a major rant, but let me first make two points by analogy.

The money analogy. Textbooks and Wikipedia say that money is a medium of exchange and a store of value and a unit of account. Generally we have no difficulty understanding that the same thing can have different uses. But the usual treatment of Interpretations of Probability discusses Logical probability and Subjective probability and Frequency Interpretations and Propensity Interpretations as if these were alternate theologies of which only one could be true. But this is silly. In discussing money, the issue isn't what money is, the issue is what money is for. Analogously the issue is what probability is for, and having different ways of thinking about what it is for is a good thing, not an obstacle.

Earth, water, air, fire. While other objects may be intuitively associated with one or more of these classical elements, few people nowadays believe this leads to a meaningful classification of objects in general. Analogously, discussing aleatoric and epistemic uncertainty via iconic simple examples of each suggests that the writer believes it is practical and useful to decompose a typical instance of uncertainty as some kind of mixture of these two concepts. But I think this is simply not true for most instances of serious real-world uncertainty about the future, as discussed here.

To me the fundamental question is

In what real world contexts is it both practical and useful to attempt to estimate numerical probabilities?

I don't claim to have a good answer, but will suggest ways of thinking about this question. One background desideratum would be an exhaustive list of contexts where we perceive chance, and the link goes to my draft attempt at compiling such a list. Another way is to ask if there are general reasons why we might care about probabilities ; this obviously relales to the useful aspect of our fundamental question. Below are three other background thoughts.

Likely/unlikely as a primitive concept

What are the key differences between human intelligence and animal intelligence? This may never have a definitive answer, but "conscious planning for the future" is surely one key difference. Wondering in what contexts humans first made conscious plans can only be speculation, though "where to search for food today" comes naturally to mind. If a creature were not aware that whether it would catch a prey animal or find plentiful edible fruit today was uncertain, we would be reluctant to classify that creature as human. And planning in the face of uncertainty requires some notion of what is likely or unlikely to happen.

A qualitative sense of likelihood, for instance a conscious recognition of some future events as likely and some as unlikely, is part of the common sense that the human species is endowed with.

Likelihood as a qualitative spectrum

There are many aspects of the world which we habitually compare on some "lesser to greater" spectrum, and sometimes we devise a numerical scale of "ratings". Here are three examples.

There is a NRS-11 pain scale with ratings of pain from 0 to 10.
Movie ratings are often given on a scale of 1 to 10, for instance on IMDb.
Crime: one can use average length of prison sentences as indicators of seriousness of crimes.

Whether one should regard such "ratings" as quantitative rather than qualitative depends, in my opinion, on whether one judges, in these examples, that

A pain rated 6 is twice as painful as a pain rated 3
A movie rated 6 is twice as good as one rated 3
A crime for which a conviction typically gets 6 years in prison is twice as serious as one getting 3 years.

I would answer "no" in these cases; the numbers should be regarded merely as a code for some verbal description (like the Michelin star restaurant ratings -- "worth a detour" etc). This contrasts with the quantitative case: saying one event has probability 60% while another has probability 30% is definitely saying that the former is twice as likely as the latter.

Somewhat bizarrely, such ratings have even been used when asking for expert probability forecasts -- see this graphic from the 2016 Global Risks Landscape in which participants were asked to assess likelihood on a scale of 1 to 7. Doing so precludes the retrospective analysis of accuracy which can be done in proper prediction tournaments.

Contexts where one does not measure uncertainty via numerical probabilities

Summary

Implicit in the usual mathematical setup of probability and statistics is the notion that one should associate numerical probabilities to uncertain events. But our examples above make the point

Whenever we think about probabilities, we are consciously recognizing unpredictability or uncertainty. But not conversely. There are many settings where we recognize unpredictability but do not naturally think in terms of chance. And there are many settings where we do think in terms of likely/unlikely but do not care to attempt a quantitative assessment of probability.

So all this is background for what I regard as the fundamental conceptual question

In what real world contexts is it both practical and useful to attempt to estimate numerical probabilities?