Although I agree, let's recall that "being intuitive" is quite a relative matter. In the present case, I find that the Carathéodory's settlement is optimal: maximal effect, minimal effort; maximal generality, minimal structure. For whom approaches the subject, and finds it not enough intuitive, I would just say (in the spirit of the celebrated motto by D'Alembert "go ahead and the faith will follow"): It's a good opportunity to train your intuition and, also, to learn some elementary techniques. Measure theory, in its elementary part, is mostly a matter of   "$ \epsilon\,   2^{-n} $"
(if you know what I mean). Finally, to go even more into the details of the construction, I would recommend to prove that the definition of measurability à la Carathéodory actually comes out characterizing the larger $\sigma$-algebra where an outer measure restricts to a measure. This makes it less out-of-the-hat, if not immediately intuitive.