The diatonic step is not a unit. Units are, by definition, uniform.

Diatonic music is written such that the intervals are treated as equivalent. But they are not in fact equivalent.

If they're treated as equivalent within the music that it primarily serves, then shouldn't our notation convey that treatment? Isn't that more important for communicative purposes than any of this useless equivocation about what the intervals actually are? If we are talking about fully chromatic music, such as the music of Schoenberg, then we agree. But within diatonic music, should we sacrifice the uncanny utility of diatonic notation for the sake of, what, intellectual consistency? By that logic, should we replace all language symbols with the International Phonetic Alphabet because read is spelled the same but pronounced differently in present vs. past tense?

Treating them as such on the page requires (in a sense, I'm not saying that competent players consciously think about this) an extra step on the instrument to undo to the convention. You have to know that a particular interval is only one fret apart instead of two.

Again, that depends on the instrument, it doesn't for the voice, it doesn't for the keyboard. Valved brass instruments are weird and make little since no matter what notation you use. And speaking of consistency of steps, the uniform move on the chromatic step is not a uniform move for unfretted string instruments, the higher you go, the closer the intervals get. So maybe we should actually have a notation that compresses as one ascends? Since that makes the most sense for those instruments. But that representation wouldn't make sense for keyboardists and vocalists don't have the range enough for it to matter, and it also doesn't make sense for the logarithmic conception of pitch space.

Who do you want this to make sense to? For what purpose will it be used? If diatonic music uses intervals in a certain way, why would we want a notation for diatonic music that doesn't show that common use?

It is just simply not true that a chromatic staff is "just as arbitrary." The conventional system describes every key (other than C major and its equivalents for the other modes) as a deviation from C major, which is arbitrary.

This is true, but so what? If this is all the chromatic staff gives us, it's faults for communicating how diatonic music works far outweigh it's benefits.