After looking at the post again and what you said I figured out what you meant. It’s true that Aug. 6th chords have a complex relationship to secondary dominants but not without some help. That help coming from tri-tone substitution. Connecting the dots and really thinking hard, a little substitution can make the relationship between Augmented 6th chords and dominant 7th’s possible. The German augmented sixth chord could be a tri-tone substitution of the (V7 of V). In the key of C major, the V7 of V works out to be D7. The Augmented 6th’s on the bVI chord would be Ab, and the tri-tone sub would be D7. Since the Italian is pretty much the same, the same works. Even the French Augmented Sixth can be rooted in a tri-tone substitution of a secondary dominant. The F6 being the tri-tone substitution of (V7b5 of V). As for resolve by theory rules, Augmented Sixth resolve doesn’t share the same resolve as Dominants. Augmented 6th’s resolve in half steps or minor sixths down to root. However, dominants tend to move in major seconds or perfect intervals. Something that is true is Secondary dominants do tend to resolve to the V of the prevailing key, and one of the common movements of a Augmented 6th is to a V chord. I’m done for awhile.