I think you are misunderstanding what the root of the augmented sixth chord actually is.
Enharmonically the German augmented sixth chord seems to be a root position dominant seventh chord. By this perspective you are thinking the root is
Ab and then there is a minor seventh above it at
Gb. Your full spelling is then
Ab C Eb Gb which is root position chord.
The problem with that is where is the augmented sixth?
In actuality the augmented sixth is from
F#! This gives us the correct spelling of
Ab C Eb F# which is a first inversion seventh chord with one very important additional characteristic_: the root is chromatically altered!
Rather than the chord being rooted on
Ab it is rooted on
F being the subdominant degree in the key of
C, it's actually a chromatically altered subdominant chord.
If you write that as a Roman numeral analysis symbol it should be
While that augmented sixth chord as the sonority of a dominant seventh chord it doesn't resolve like a dominant seventh.
With an actual dominant seventh in the key of
C the root of the
V7 (G7) chord will be
G and it will be held as a common tone when resolving to
In the case of the augmented sixth chord the alter fourth - the
F# - is not held and has a strong pull to resolve up to the dominant scale degree
G. That particular movement makes clear the subdominant function of augmented sixth chords. It resolves to the dominant.
...the same interval as the bVI7
C minor the chord would be
Ab C Eb G with a
G natural which makes a major seventh chord. That different should help underscore that the German augmented sixth chord is not a type of
You may be interested to know there is another augmented sixth chord spelled
Ab C D F#. Called the French augmented sixth chord.
The root of that chord is
D which makes is a chromatically altered supertonic seventh chord. You could use the Roman numeral symbol
ii#6/4/3 for it.
In the jazz world both of these chords can appear in enharmonic respellings as tritone substitutions.
The German augmented sixth as a tritone substitution could look like
bII7 as in
Dm7 Db7 C6.
The French augmented sixth as a tritone substitution could look like
bII7b5 as in
Dm7 Db7b5 C6.
In that jazz context notice that the chords are substituting the dominant chord
V and that would be the rationale for why they are enharmonically respelled and don't fulfill the subdominant function of the alternate spellings of the German or French augmented sixth chords.
The important take away point is: enharmonic spellings should reflect how the chord is functioning!