In the example, you agreed to assume the coin is fair. That is, you agreed to stick to the model even if it turns out to be wrong. Throwing the model away, however suggested by the evidence, is just violating the rules you agreed to.
Or maybe to be nicer about it the proper response is "So are we talking about the real world because in the real world the odds of 99 heads in a row are 1 in 2^100 which is effectively zero so I just want to verify you understand that your hypothetical situation and conclusion will have no basis in reality. If you actually saw 99 heads in a row then the odds are infinitely higher the coin is broken and is not fair"
A) a hypothetical introduced by the idiom "Assume that..." (which introduces a statement of fact in the hypothetical situation), and we are discussing a scenario in which the coin is in fact fair, in which case Dr. John is correct and Fat Tony is committing the gambler's fallacy, or
B) a scenario in which a third-party tells someone that the coin is fair, but that is simply an assumption that should be updated by new evidence, in which case Dr. John would be committing the ludic fallacy.
The example isn't instructive because anyone reading this and thinking of scenario A isn't disputing the idea that, if this happened in reality, you should throw away the model. They're just interpreting the phrase "assume that" differently.
The fact that your model can be justified as technically correct is irrelevant. When the data doesn't fit the model any more, the supposed absolute correctness of the textbook model blocks your ability to understand what's happening.
This isn't even about games, but about whether it's possible to be open-minded enough to consider that simple textbook "solutions" are not guaranteed to be a good fit for real problems - especially if you don't understand their limitations or the assumptions they're based on.
And it doesn't just apply to statistics.