I keep seeing around the place that chord names shouldn't describe the voicing of a chord, but simply the intervals. If this is the case, why do the intervals ♭13, ♯11 and ♯9 exist? Are they not just the equivalent of ♯5, ♭5 and ♭3?
Names such as ♭13, ♯11, ♯9 indicate "extensions" of the chords, on top of the basic triad (or seventh, as the case may be)
So for example C7♯9 means that you have a basic major triad (C-E-G) and a dominant 7th (B♭). And on top of that, you have a ♯9, which is an octave above a ♯2, and that's a D♯. And that D♯ is marked as a ♯9, because D♯ is not there to play the role of a minor third, which would make the chord a minor chord, but is a "color note" on top of the dominant 7th chord.
One more thing to keep in mind: those extensions are usually voiced at the top of the chord. For example, in the "Hendrix chord", e.g. E7♯9, that ♯9 is always the highest note of the chord. If you played that ♯9 next to the major 3rd at the bottom of the chord, you'd get a super-dissonant clash between major 3rd and minor 3rd (i.e. augmented 2nd, i.e. ♯9) intervals. But if you leave the major 3rd at the bottom, and the ♯9 at the top, it sounds great.
You can make a similar analysis with all the other chord extensions you mentioned.
...why do the intervals b13, #11 and #9 exist? Are they not just the equivalent of #5, b5 and b3?
There are existing questions about this, but it may be hard for you to find them.
Intervals above the root higher than an octave (for practical purposed intervals bigger than a seventh) are called chord extensions. So,
b13 #11 #9 are chord extensions, but
#5 b5 b3 are not, because of size larger or smaller than an octave.
Side note, music theory distinguished between these interval categories as simple versus compound intervals. Ex, a simple interval is a major third, but adding an octave to it makes the compound interval of a major tenth.
When you have chord symbols with extensions/compound intervals - like
G9 - by convention the chord is understood to contain a seventh. Written out explicitly the chord is
9 is an
A natural over
G7 which is
G B D F.
9 was the simple interval
2, the chord is a basic
G B D with an added
2 and importantly no seventh! It's a
Gadd2. That's mostly a theoretical example. Is more common to have a
Gadd9 name. The
add means add the interval - the ninth in this case - to a basic triad with no other assumed intervals.
Gadd9 then is a major ninth added to a
G major triad
G B D A.
Finally, the big distinction between
G9 is a dominant chord but the
add chords are not.
b13 #11 #9 are used it means the base chords are seventh chords (and unless they are marked otherwise with
min or something else the chords will be dominants.)
When simple intervals like
#5 b5 b3 are used the chords aren't seventh chords.
One quibble is
b3. Using that doesn't seem to be necessary, because you would just label the chord
Gb3 would normally be written
Gmin. If you wrote something like
G7b3, it's confusing. Is it supposed to mean
G7#9? If the meaning is to use a sharp ninth, then write that. I think most people like to include the seventh
G7#9. If the naming system was very standardized you shouldn't need to write the
7, but the fact is the system isn't completely standardized.
How could they possibly be? D4 isn't the same as M3, so what would make ♭13 the same as ♯5 (which should be augmented 5).
Maybe what's been missed is that there are at least two different names for any one sounding interval. For technical reasons.
But getting back to your ♯9, for example. It comes after the octave ( it's >8) so already in a ♯9 chord, there's 1,3,5,♭7. Let's put it into key: C E G B♭. Then on top, almost an octave above that E, there's D ♯. It might sound like E♭, but if your idea was employed, it would be played right next to the E. Try it and listen!
No, the numbers are larger than 8 because they are played above the basic triad plus root octave (if needed). Were they as you suggest, some of those chords would just sound muddy. On any instrument. Try it.
All that apart, chord tones normally work using the odd numbered notes (6ths aside!), so a convention already exists, and since most chords will have a 5 anyway, it is less confusing to not duplicate a note name.
And you're right - chord names don't reflect the voicings - not even slash chords (but maybe power chords..!)- but merely the note make up.
Chord symbols are a naming convention for describing chords that consist of stacks of thirds. As an exception to this basic form there are "add" chords where you simply add a note that comes from outside the stack of thirds. A more systematic name for a "6" chord would be "add6", but to keep it shorter it's called just "6". Another exception or modification to the basic form are "sus" chords which "suspend" the third by substituting it with another note. (think about suspenders! elastic strap pulling the chord's third to some direction)
Even though the higher-than-octave notes in the 9, 11 and 13 chords you're referring to could be mapped back to the first octave, they're looked at as stacks of thirds in the culture for which chord symbols were invented. There can be different voicings where some chord tones are spread to octaves (or duplicated) in various ways, but the normalized stereotypical form of a 9 chord is a stack of four thirds on top of each other. Not a "7add2". It's a cultural convention. (There are reasons and cultural history behind this convention though, it's not arbitrary)
If there was a musical culture that doesn't look at chords as stacks of thirds but by placing all notes in the same octave, then for that hypothetical culture, you would probably have different meanings and conventions for chord symbols.