What am I misunderstanding about this constructive proof that $\mu(\mathbb{Q}) = 0$?

What this example tells you is that your intuition about these things is not really reliable. Don't worry too much about that; everybody goes through it.

In particular, the union of the intervals does not cover the entire real line. There are gaps -- they are small (none of them contain an interval), but there are a lot of them, and somehow they manage to add up to something with positive measure.

I get your confusion, but a set having "gaps" does not imply the set misses an open interval. For more trivial examples, consider $\mathbb R\setminus\{\sqrt{2}\}$ or $\mathbb R\setminus\mathbb Q$. The first set has a single gap but still contains every rational number. The second set has gaps that are dense in the whole real line, but have zero measure. The cover that has been constructed in your question covers every rational number (and hence many irrational numbers), but misses "most" of $\mathbb R$, as evidenced by the fact it has very small measure.