I think the important thing to consider is: what's your purpose?
Music is a language, in that it expresses through sound. Music notation is another language, one that represents that expression as an analog through symbols. And music theory is still another (and separate) language that abstracts the ideas of music through terms.
When a language abstracts something, details get lost. You tell me you just bought a car. "Car" is an abstraction that gives me a general idea of what you just got - I know you didn't buy a birdcage or a canoe. I know the thing you bought probably has four wheels and you can use it for transportation. But I don't know if it's a Ford or a Fiat, or if it's green or yellow, because abstractions sacrifice details for efficiency.
Chord names are an abstraction. You tell me it's a C7 chord, and I know the tones are C, E, Bb, and maybe G. Or maybe not G - the "open" C7 chord on a guitar usually leaves that note out, as do common piano voicings for one hand.
You could be more specific, and tell me you bought a blue car. Or that you want me to play a C7/E chord. Or give me a figured bass E with 65 written above it. But I still don't know if your blue car has two doors or four, or if the C7 will have the Bb in the alto or soprano. We have different levels of specificity when we abstract ideas.
If you want to tell me exactly what pitches make up a chord, the tool to use would be standard notation. But even that is an abstraction - I'm primarily a guitarist, and for many written chords I can achieve the same voicing of pitches in more than one way on the instrument, and each way will have a slightly different timbre.
I don't see any real advantage to specifying the order of tones in a chord symbol, because chord symbols are a fairly low-level of abstraction. They're designed to get a very general idea of the harmony across to another musician, in the same way that a script for a play tells an actor what to say, but not how. If the script says "You're going" it's up to the actor to flesh it out - is she surprised that you're going? Delighted? Angry? Commanding?
There are lots of ways to deliver a general instruction. I'm not sure that complicating chord symbols will add as much as you expect, because it's still an abstraction - and we already have better tools available for communicating more specific instructions.