If you know I, ii, iii, IV, V, vi in C then you'll be able to play tunes in A natural minor. Add III to cover A harmonic minor, and II to cover A melodic minor
In real life though the key usually depends on the singer's vocal range, so you won't really get away with learning just one key
In A harmonic minor the III is already a C, you can make it augmented but that is not very common. You generally change the v to a V and keep the VII diminished in both cases where as it is major in the natural minor.
He was saying that the III of C major is the V of A minor, and so if you are planning to play in A minor using a set of chords pulled from C major, you may want to add a III alongside the iii.
Yes, that's a good point in the case of six keys (I think my point holds for just learning three), but it is not the one made that the article seemed to make.
In real life though the key usually depends on the singer's vocal range, so you won't really get away with learning just one key