If you see it from a pure classical standpoint, a 6th can never be a part of a chord-signature. Every chord is built by adding another 3rd to its predecessor note - starting from the root note. So you only get 1 3 5 7 9 11 13...
Even a chord like e.g. C6 is interpreted as a minor 7th chord with the 3rd of the chord as root -> A C E G (1 3 5 7) -> C E G A (3 5 7 1).
In modern chord theories you'll find these naming conventions that allow a e.g. Cm6 to be seen as 1 b3 5 6. But also here you get a limitation - being that only if there is no 7th included this 6th may actually be written out as 6th.
So there can be a C 6/9 or a Cm6/9 but if a 7th is included it has to be written as 7/13 or simply 13. Every tension bigger than 7 (this being 9 11 13) does automatically include the 7th, so there's no need for writing it explicitly into the signature.
You also see that this explicitly written 6th in a 6/9 chord contradicts the rule that every tension bigger than 7 includes automatically the 7th. There is NO 7th in a 6/9 chord.
And of course in all these examples I am talking about the flat seventh 'b7'. Otherwise there had to be an indicator maj7 to tell that we are not talking about the flat one...
So this 6/9 chords do always have TONICA functionality (root) and not DOMINANT functionality!
Of course you will find chords like e.g. C4/7 which may also seem to be a contradiction to the above said. But this 4th isn't actually a tension like 7 9 11 13 it is rather a Csus4/7 chord which means at some place it will get resolved to the 3rd -> sus4-3. The same thing happens to a sus2 chord which will very likely become a sus2-3 later on. So you do not want to write these chords as 9th respectively 11th chords because they do not include the 7th. They rather omit the 3rd by suspending it with the 2nd or 4th. Thus the notation says 'sus'.
That is why you will never find a 11th chord but rather a sharp 11th (#11). Here the tension is two half steps away from the third and doesn't interfere. Every 11th is very likely to be an incorrect notation of a meant-to-be sus4 chord. And also chords (very typical for 80's style american pop music like Barbra Streisand and the like) like C 9/#11/13 are incorrect notations, because what they typically want to achieve is C - D/C... - a very nice progression - but better written down like this! Because the notation C 9/#11/13 would automatically include the 7th which would totally break the charming character of the C - D/C progression, especially when the thirds are the highest notes in the two chords...
(G C E over the root of C -> A D F# over the root of C).
The scale over this progression is somewhat LYDIAN like you can hear it in almost any work of the grand-master Lenny Bernstein -> MA-RI-A from Westside Story or the OUVERTURE from the Musical Candide. Also Ennio Morricone uses it a lot (Harmonica in 'Once Upon A Time In The West' etc.)
The last thing to mention here is a phenomena like you will find it in e.g. John Lennon's song 'Imagine'. Here the right hand of the piano changes between a 'D E G' (2 3 5) chord over a root of C and a 'C E G' (1 3 5) chord.
This is no sus2 chord because the 3rd is already there (E). So this is an additional note for the chord an has to be written as add2 or add9. This add prefix is simply saying that there is no 7th automatically included by tensions bigger than 7 (9 11 13) and that the 3rd is not omitted like in sus chords. And to that it resolves its intrinsic tension to the root note of the chord D-C-D-C-D-C...
So you will most likely have no root note or octave of the root close beside this add9 (because it acts almost like a sus chord - but going from 9-8 -> add9-8. (So you could definitely call it a sus9 chord -> sus9-8 ;-)
But I personally like it better and find it more correct when it is written out as add9 instead of add2 because that indicates much better the stretched rubber band effect that the 9th causes - and wants itself to get relieved by lowering this tension to the root (8th / 1st).
So as to your question - a flat 6th will most likely be a flat 13th (b13) in a 9/b13 or a 7/b13 (b13) chord and have DOMINANT functionality!
As a sharp 5th (#5) it can have both - TONICA (root) and DOMINANT functionality. But this #5th has to be raised one half-step further in the following chord [D# - E] . This leads to a 6th if the root stays the same (some kind of TONICA functionality)
G #5 -> G6
(G B D# -> G B E both over a root of G)
or (when the 7th is included) to the 3rd of the TONICA (root) chord
G 7/#5 -> C
(G B D# F over the root of G -> G C E over a root of C)
In this case the G 7/#5 has a DOMINANT functionality!
And - of course - when it gets jazzy a DOMINANT chord like
(A B D# F over a root of G)
can advance to
(G B D E over a root of C)
Here the tension of the 7th (F) becomes the 3rd (E) of the root chord, the #5th (D#) DESCENDS to the 9th (D) of the root chord and the 3rd (B) stays on its place to form the maj7 of the root chord Cmaj9...
Hope that helps!