Let's start w/the names: Italian, German, & French. These names have nothing to do with where these chords came from, there is no heritage attached to these chords, there just names. Italian composers used German 6th chords, as well as polish composers and Russian composers, so stop trying to blame the French, Italians, and Germans for creating some stupid form of theory, Musicians of the Eoropean nations created these chords and the theory behind them.
Secondly, these chords are for compositional purposes. The rules for the usage,Identification, and resolution are very explicit and would be hard to explain in a single paragraph, you really should research this topic on your own, just for your own peice of mind. Like I said before, these chords are named "augmented 6th " for the chromatic interval that is created from the voice leading. That is where the misconception lies, becuase they are essentially Dominant 7th chords when anylized in a vertical fashion. But since the Dominant 7th chord was not yet recognized yet as a individual sound form they named them for the voice leading technique, because thats what they where utilizing these chords for anyway
Ex: Write out a iv-7 chord in the key of A- in 1st inversion, use this voicing (Bottom up) F C D A. Now write out a A- in 2nd inversion using this voicing; E C E A, now if you inserted a chord in between these two chords to connect them chromatically you would have; F C D# A. That is how the classical augment 6th chords came into existence. They were named after the interval created between the bass note F and the 3rd voice D# when chromatic voice leading was implied.