Minor chords with inner movement within the harmony are very common in different styles of music. They may have a specific name or history, I can’t recall offhand. When a chord progression works it tends to get used by many different artists, possibly with variations, and sounds familiar to us.
The reason Am to Am6 to G has the triumphant and conclusive nature you described is very simple. Those 3 chords basically function in a very similar way to a ii-V7-I, a subdominant-dominant-tonic resolution. After the Am you have an Am6, which is basically a D7 with an E instead of a D in the chord (or a rootless D9/A). It also is the first inversion of F#m7b5, which is a viio chord, another dominant chord in G major. That chord has most of the defining elements you need for a V7-I resolution, which is the F#-C tritone resolving inward to the G-B root and third of the I chord. The only thing it lacks is the descending 5th movement from D-G in the bass.