I don't like your use of the word "probability". Sometimes, you use it to describe subjective probabilities, but sometimes you use it to describe the frequency properties of putting a coin in a given box.

When you say, "The brown box has 45 holes open, so it has probability p=0.45 of returning two coins." you are really saying that knowing that I have the brown box in front of me, and I put a coin in it, I would assign a 0.45 probability of that coin yielding 2 coins. And, as far as I know, the coin tosses are all independent: no amount of coin toss would ever tell me anything about the next coin toss. Simply put, a box, along with the way we toss coins in it has rather definite frequency properties.

Then you talk about "assigning probabilities to each possible probability between 0 and 1". What you really wanted to say is assigning a probability distribution over the possible frequency properties.

I know it sounds pedantic, but I cringe every time someone talks about "probabilities" being some properties of a real object out there in the territory (like amplitudes in QM). Probability is in the mind. Using the word any other way is confusing.