According to this summary: https://andrewdiruzza.blogspot.com/2011/02/pat-martino.html
... the augmented triads are used as a "practical aid in visualizing twelve diatonic keys on the guitar in a horizontally smaller area on the instrument." You're not supposed to play the A augmented (if you don't happen to want an augmented chord). You're supposed to see the shape on the fretboard, and the minor triad as a modification to the augmented shape. This helps you locate and create chord voicings in different positions on the fretboard. You don't have to know all possible major and minor voicings as such - if you can see the augmented shapes, you can create major and minor chords from them.
Compare this with the black and white key arrangement on the piano - all keyboard players use the black and white keys as a visualization and location aid without even thinking about it, because it's built into the instrument and it cannot be removed. If the piano keys were all the same color and if they were only side by side chromatically without separation to a C major scale and "sharp/flat keys", ... and if it wasn't linear but the same pitch in many different places, then you would have to invent some kind of a system to find your way around the keys and to be able to operate. Like what you have to do if you're a guitar player. :) As far as I understand, with Pat Martino's method you build a (supposedly) more guitar-friendly visualization aid into your mind, by training yourself to see minor and major chords as modifications of augmented triads.
For example, here are two ways to obtain the Dm chord from an A augmented triad. The augmented triad is marked with black dots. I guess Pat Martino sees dot patterns like this on the guitar fretboard.
Will this system of looking at chords and the fretboard help you? Maybe, but it will take practice. According to a post on the Yahoo Jazz Guitar Group (YJGG) discussion thread archived by Google, it takes years to really master this skill