The fundamental conceptual question about probability is …

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 xxx is to ask if there are general reasons (xxx add link to new page) 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. Whether one should regard such "ratings" as quantitative rather than qualitative depends, in my opinion, on whether one judges, in these examples, that 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 probability

Clearly one can go through much of ordinary life with only common sense notions of likely/unlikely, illustrated by the saying "When you hear hoofbeats, think of horses, not zebras". A noteworthy context in which non-quantitative assessment is mandated is the famous "beyond reasonable doubt" criterion for criminal conviction. The legal profession explicitly refuses to quantify this; if you as a juror were to ask the judge whether a 97% probability was sufficient, the judge would not give you a straight answer!

As a more substantial example, the Intergovernmental Panel on Climate Change (IPCC) issues periodic reports, widely regarded as the most authoritative analysis of scientific understanding of climate change caused by human activity. Future predictions involve uncertainty, and they want their many authors to be consistent in how they write about uncertainty, so provide technical documents such as Guidance Notes for Lead Authors of the IPCC Fourth Assessment Report on Addressing Uncertainties from which I have extracted the table below, there labelled "A simple typology of uncertainties".

Type Indicative examples of sources Typical approaches or considerations
Unpredictability Projections of human behaviour not easily amenable to prediction (e.g. evolution of political systems). Chaotic components of complex systems. Use of scenarios spanning a plausible range, clearly stating assumptions, limits considered, and subjective judgments. Ranges from ensembles of model runs.
Structural uncertainty Inadequate models, incomplete or competing conceptual frameworks, lack of agreement on model structure, ambiguous system boundaries or definitions, significant processes or relationships wrongly specified or not considered. Specify assumptions and system definitions clearly, compare models with observations for a range of conditions, assess maturity of the underlying science and degree to which understanding is based on fundamental concepts tested in other areas.
Value uncertainty Missing, inaccurate or non-representative data, inappropriate spatial or temporal resolution, poorly known or changing model parameters. Analysis of statistical properties of sets of values (observations, model ensemble results, etc); bootstrap and hierarchical statistical tests; comparison of models with observations.

This table is addressing the issue of uncertainty and mathematical modeling. It makes the point that, within a complex setting (such as future climate change), any asserted numerical probability is (at best) an output from some complicated model in which all these different kinds of uncertainty are present. This point is obvious once you think about it; but it's just different from what's said in textbooks on the mathematics or philosophy of probability.

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?