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The Univariate Fallacy

I agree with your other two points (mostly - I don't feel that the distinction between discrete and continuous variables is important for Zack's argument so it seems fine to gloss over it) but I disagree with the first.

In order to be able to simply multiply likelihood ratios, the sufficient fact is that they're statistically independent. In this toy model, they also happen to be identically distributed, but I think it's clear from context that Zack would like to apply his argument to a variety of situations where the different dimensions have different distributions. You're suggesting replacing "X, therefore Z" with "X and Y, therefore Z", when in fact X->Z, and it is not the case that Y->Z.