The main reason why "flatted 10th" is a less useful designation for a chord name or chord symbol is because it violates the very precept that chord names/symbols were invented for:
Chord names/symbols were developed to be a form of shorthand that can easily convey the constituent pitch class components of a simultaneity without regard to register, and, at least in the post-figured bass world (aka, modern usage), without regard to voicing.
"C7#9" is completely register- and voicing-agnostic. There is nothing about that chord as written (there) to indicate what octaves any of the chord tones occur in, or how many times any of those chord tones could/should be doubled in different octaves, or what inversion it is played in. That's the point. If you wanted to be more specific about the register and voicing, you would write out a vertical simultaneity in standard musical notation.
Conversely, "C7b10" [sic] very specifically indicates that the note Eb is to be played in the register one octave above the root note C. I suppose that might be handy if that's exactly where you want the Eb to sound...but then what register should the Bb or the E natural be played? The "flatted 10th" chord symbol doesn't specify...so it's both incomplete, and unnecessarily restrictive at the same time.
