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The problem is that this doesn't get at many of the intuitive properties of entropy.

A different explanation (based on macro- and micro-states) makes it intuitively obvious why entropy is non-decreasing with time or, with a little more depth, what entropy has to do with temperature.

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That doesn't strike me as a problem. Definitions are often highly abstract and counterintuitive, with much study required to understand at an intuitive level what motivates them. Rigour and intuition are often competing concerns, and I think definitions should favour the former. The definition of compactness in topology, or indeed just the definition of a topological space, are examples of this - at face value, they're bizarre. You have to muck around a fair bit to understand why they cut so brilliantly to the heart of the thing.

The above evidently only suffices as a definition, not as an entire course. My point was just that I don't think any other introduction beats this one, especially for a book with the given title.

In particular it has always been my starting point whenever I introduce (the entropy of) macro- and micro-states in my statistical physics course.

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