The point of the not quite fair die example was to demonstrate where 'probabilities' are coming from. The fair die, after several bounces, maps the initial state space into the final side-up states in a particular way, so that 1/6th of even a very tiny part (hypervolume) of initial state maps to each side-up final state. The not totally fair die is somewhat biased from that. Any problems involving die can be solved from first principles all the way from this through selection of the parts of initial state that are compatible with observation, to the answer.
With regard to the statisticians not losing sleep over that, there is a zillion examples in practice where you have to deal with e.g. electric current, or temperature, or illumination, or any other fundamentally statistical property, and you have limited computational power. A lot of my work is for doing this on illumination; I have to compute illumination in a huge number of points on the screen (and no you can't bruteforce even if you had 1000x the computing power, not to mention that when there's 1000x the power you'll have tighter constraints on error and time). I don't really care if some people don't find anything wrong with doing a wrong thing "because we won't be beaten in practice", when I am earning some of my money by beating those folks in practice. So better for me that some folks just don't understand that you shouldn't get to choose some arbitrary numbers. Yes, in various really fuzzy problems, you can do what ever you subjectively please. But to see this as fundamental - that's quite seriously silly.
There are many methods for finding out the resulting distribution; one particular method involves more regular sampling of the initial state than random (e.g. grid with jittering), so that you get error that improves much better than 1/sqrt(N) ; it can in principle be used for die simulation, and is used in practice in similar problems that are less messy (molecular dynamics comes to mind) . I generally find that nowadays a lot of very important insights are within the more applied fields; the knowledge has not yet propagated into this meta-ish land of arguing mostly over terminology and not having to be maximally correct against golden standard of reality.
Any problems involving die can be solved from first principles all the way from this through selection of the parts of initial state that are compatible with observation, to the answer.
You're sketching out a methodology for solving forward problems (given model, determine observations), which is fine but it's not what motivates statisticians. Statisticians are generally concerned with the backward/inverse problem (given observations, determine model).
In reality, we're not presented with complete and accurate technical specifications for the die/table/th...
I've had a bit of success with getting people to understand Bayesianism at parties and such, and I'm posting this thought experiment that I came up with to see if it can be improved or if an entirely different thought experiment would be grasped more intuitively in that context:
I originally came up with this idea to explain falsifiability which is why I didn't go with say the example in the better article on Bayesianism (i.e. any other number besides a 3 rolled refutes the possibility that the trick die was picked) and having a hypothesis that explains too much contradictory data, so eventually I increase the sides that the die has (like a hypothetical 50-sided die), the different types of die in the jar (100-sided, 6-sided, trick die), and different distributions of die in the jar (90% of the die are 200-sided but a 3 is rolled, etc.). Again, I've been discussing this at parties where alcohol is flowing and cognition is impaired yet people understand it, so I figure if it works there then it can be understood intuitively by many people.