I think this needs to be answered from the conventional point of view rather than logical, because jazz chords symbols provide a logical system, but conventionally people don't always strictly follow the system.
A good example is dominant ninth chords.
G9 would be a ninth chord and the seventh is understood to be in the chord despite it not being written out. I think that is mostly an issue of visual formatting. If you just wrote
G79 it looks funny... a seventy ninth chord?!?.
G7(9) is cumbersome, so the shortened
G9 is used. Then, for some reason, altered ninth chord will include the
G7♭9. There doesn't seem to be a logical reason to not simply write
G(♭9) to clarify the flat. It seems part of the convention is to avoid writing parenthesis. Drop the
7 to avoid parenthesis, but other times include the
7 to avoid parenthesis.
G#11 should be understood as a
G dominant seventh chord with a sharp eleventh added. But it seems the conventional symbol is
I think the reason is because the
#11 is associated with lydian and that brings a different tonality than major/minor with blue notes.
If you look for the diatonic chord, in major, that has diatonic sharp eleventh, an augmented fourth, it's the
IV chord. In
C major that would be
F A C E G B, with the
B being the "sharp eleventh." But notice that the seventh is major. To write that in jazz you need to specify the seventh:
FΔ7♯11. That is a lydian chord.
There is also the lydian dominant chord & scale which is a form of the melodic minor scale. That scale has its own special tonal identity. Conventionally the chord symbol, for an
F root, is
F7♯11. Logically you would drop the
7 and just write
F♯11. But I think the convention to include the seventh is probably to make the distinction between the two important lydian types.
I don't think the same kind of logic versus convention comes up with
13 chords simply because
♭13, because a sharp thirteenth is enharmonically a minor seventh, and a flat thirteenth ends up sounding like a different root, i.e.
Cm7♭13 sounds more convincing as the inverted