My question is about the notation of chords and what that notation conveys (mostly applied to jazz and the notes that are implied by the chord symbol). A chord symbol notated with a '7' alone indicates a particular quality for the 3rd, 5th and 7th. So if someone writes CM7, then we know that the 3rd is major (and not minor, augmented or diminished), the 5th is perfect (and not augmented or diminished), and the 7th is major (and not minor, augmented, diminished, double diminished). The existence of a perfect 5th eliminates the existence of an augmented 5th, and vice versa. It doesn't make sense to convey a major augmented ♯5 mood by writing a CM7 chord, because the CM7 chord symbol indicates that the 5th is not augmented. However, since the CM7 symbol doesn't specify the quality of the 2nd, 4th and 6th, it could convey a lydian, jonian, harmonic major mood, and others. Is this reasoning correct?
If this reasoning is correct, CM7(♭5) cannot express a lydian intention because the chord notation would imply the absence of a perfect 5th and a lydian mode contains a perfect 5th. So if the intention is a lydian feel, then the appropriate chord would be: CM7(♯11) (being aware of the 9th, which usually is not a problem).
Likewise, G7(♯5) couldn't express a phrygian dominant mood because the phrygian dominant mode contains the ♮5. If the intention is for the soloist to play phrygian dominant, then the chord should be written as G7(♭13), thus allowing the existence of a perfect 5th alongside the ♭13. (I know the 5th is usually omitted when voicing this chord, but that's not something the chord notation is formally telling us to do; it should be a matter of taste, not obligatory. Even if the pianist/guitarist omits the 5th when voicing the G7(♭13) chord, the chord symbol nevertheless implies what notes are available to play.)
In practice it seems to me that this information, embedded in the chord symbols, is overlooked and G7(♯5) is treated as the same chord--with the same implications--as G7(♭13). The idea here is that different enharmonic spellings imply different available notes (and therefore scales), particularly in regards to alterations like ♯5 vs. ♭13 and ♯11 vs. ♭5. This idea could only be implemented when dealing with the known diatonic scales and their modes, because one can point easily to scales with two types of 5ths or 7ths, etc. Is this something to pay attention when reading charts or is there no hope in this level of formalism?