However, it is notable that it works much better in some constructions rather than others, and that while it might trip off the tongue in a situation where the gender is unknown or the number is irrelevant such as "If someone comes in, tell them I am busy" but it does not do so easily in the examples Squeegee gave where number becomes confusing.
That's right, and that is mostly because it is already in common use, sounds natural (in some contexts) and has been for a long time.
But it is difficult to overhaul very common words in the English language in a way that gains widespread acceptance. Titles such as Ms, are fairly low-frequency words so they are not a major mental imposition, but pronouns are used everyday by everyone, often several times in a sentence, and they come with challenges of subject-verb agreement. I mean it is bad enough for some people who have been mistaught that they need to always say "you and I" instead of the far more natural, and in many cases actually correct, "you and me". To get prescriptive about pronoun use in the way that you argue is way, way more onerous than you seem to realize!