Speaking as someone who does in fact speak, read, and write Japanese, I would like to preface this by saying that I greatly enjoy and continue to use the Hepburn system.
I think there can be two purposes to a romanization system for any given language, and more generally, any system used to transliterate from one language to another.
The first, which you're getting at, is accurate representation of the original phonetics to as precise a degree as possible. Obviously this has value in situations such as the one you point out, where two similar items have subtly different pronunciations.
The second is to be useful to a non-native speaker with possibly not even a passing familiarity with the language. I don't know about you, but I don't know what /ʧ/ is, and I surely don't expect my friends, or your run-of-the-mill travel to Japan to know it either. I'm also not a fan of accented letters like 'ô', which in a romanization, inches closer to IPA territory. Surely it's my American perspective speaking, but accented letters are, to me, devoid of information and I can't help but interpret them as noise rather than signal.
I argue that if precision is your most desired quality, then you may as well go whole-hog and just adopt IPA for your rominization needs. Linguists have figured out how to represent sounds with precision, so we may as well piggyback on their hard work.
If however, you think that a romanization should be useful rather than precise, then Hepburn makes quite a bit of sense, barring it's occasional quirks like 'wo' for 'を'. The system has traded precision for ease of use. Was some information lost in the process? Yes, absolutely. Intonation is gone, for instance. But although intonation exists in the language. it just isn't that important in Japanese.
Textbooks for other languages have gone with more precise romanization system and suffered for it. I've gone through some Korean textbooks, and they generally all use a weird combination of IPA and some uncommon (to us non-linguists) ways of romanizing hangeul, which made the initial few weeks of learning depressingly difficult. There was a clear overfocus on precision of pronunciation, but as an introductory level learner, I just didn't care for that level of accuracy. A system that is easy to understand and pronounce at first sight at the cost of some phonetic accuracy is well worth it for the casual observer. When extremely high accuracy of pronunciation becomes important to you, IPA is always around to show you how it's really supposed to sound.