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/thrower system we encounter. All we get to see is the sequence of sides that landed on top. If we're playing a game that uses the die, it's of interest to know how this sequence will continue into the future.
One general approach to figuring this out might involve inferring technical specifications. Maybe if we're really clever, we can figure out what grade of steel the die is made of just from the observed side counts. Less ambitiously, we might try to recover the relative side lengths and rounding radius. With all this information, we can then simulate forward to estimate the sequence of future throws. The number of parameters involved here may number in the tens or hundreds, or into the millions if we want to capture all the physiological details of a human thrower. It's also not quite clear whether a system like this would even converge to any stationary long-term behavior from which limiting relative frequencies could be calculated.
Another approach is ignore all the detail, assume independent identically distributed tosses, and just try to learn the five parameters (P(side 1), ..., P(side 5); P(side 6) = 1 - P(side 1) - ... - P(side 5)). Forward simulation in this case is just repeated sampling from the learned distribution.
Moreover, let's suppose that (effective) independence emerges from the technical specification model. Then we have a huge identifiability problem; all those hundreds of parameters are just providing a redundant parameterization of the iid model. We can't hope to learn all of the parameters from the data we get to observe.
I guess as long as you want to stick to forward problems, you can invoke Occam and deny that probability even exists. But don't assume that your understanding carries over to inverse problems. Probability is a useful technical tool there, and applying it to real problems requires translation/operationalization. Two different frameworks for this are frequentism and Bayesianism.
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.
If you want to put your money where your mouth is, I have a proposal. Take a die of your choosing, or manufacture one according to your own specifications; it doesn't have to be remotely fair. Also supply a plate onto which it can be tossed if you desire. Do whatever measurements you want on them. Then convey them to a mutually-accepted third party. The third party rolls the die 200 times, according to instructions you publicly post, and then publicly posts the first half of the sequence of rolls and a hash of the second half of the sequence. We both predict the side counts in the second half of the sequence and post the predictions publicly. The third party reveals the second half of the sequence (which can be checked against the hash) and whoever was closer to the true side counts (in squared error distance) wins. The loser pays the winner some mutually-accepted amount, plus or minus half the die/plate shipping expenses as appropriate to split that cost.
I am an applied mathematician who actually does work on finding the values of probabilistic quantities in better computing time than straightforward numerical experimentation. Probability is not just statistics.
In so much as what you think Bayesians do deviates from what I know has to be done, you have a wrong idea of what Bayesians do (or giving you benefit of the doubt at expense of others, are referring to some "Bayesians" whom are plain wrong), or something like that but the discussion is too fuzzy for me to tell which. (Ditto for frequentist...
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.