Note that any combination of two of the three "dimensions" does constitute an independent system. That is, given that q + r + s = 0, then q + r = -s, so you could use only (q, r) as your coordinates (and indeed that's likely how you'd canonically refer to them)
The "third coordinate" is only for convenience when working with the three main axes of an hexagonal grid. Without it, the three directions of your grid are {+q, +r, -q-r}, which introduces an ugly asymmetry for one of the three axes.
The "third coordinate" is only for convenience when working with the three main axes of an hexagonal grid. Without it, the three directions of your grid are {+q, +r, -q-r}, which introduces an ugly asymmetry for one of the three axes.