E.g. in the author's example, where the rate starts at 5% and increases by 5% with each failure, the expected number of trials needed for a success would be:

    sum = 1 * 0.05 +                // hit the first trial
          2 * 0.95 * 0.10 +         // missed first, hit second
          3 * (0.95*0.9) * 0.15 +   // missed first two, hit third
          // ..and so on

For the reverse case, there are arbitrarily many such sequences that would average out to any given value. In practice someone using this technique for gamedev wouldn't necessarily only want X+X%.

Every left child produces a tree from that point which is exactly the same at the root tree.

Every right child produces a probability of +5% until you get to 100% after which the right node is forced back to 5% — meaning it will match the root from that point.

I feel like once you’ve defined the tree to depth 20, you could average all the probabilities and it should equal 19%, because everything from that depth further is just a uniform repetition of the probability tree, but it’s probably slightly more complicated than that.

No. Probability theory is about working forward from a given model to a set of possible observations. And Bayesian statistics is about working backwards from given observations to which model in a set of models produced them (which is why an old name for Bayesian statistics is 'inverse statistics').

In this case, there is no given set of observations; there's not some game out there which has produced a dataset and you're trying to work backwards to figure out what model that game is using. Instead, you have a bunch of models and you're trying to work forwards to figure out which of them will give you the kind of outputs you want. So, that's probability theory.