There's another thing to remember, it was apparently different in TOS.
From TWOK onwards into TNG and beyond, we've seen ship battles in unusually close (and probably unrealistic) quarters. In these situations, Z Axis maneuvering would be a factor.
But in TOS, we rarely saw fighting ships in the same frame. This would accurately reflect the ranges, which were often given in tens of thousands of kilometers. This would be well beyond visual range (excluding sensor magnification).
So what I'm supposing is, if the enemy range is 15,000 kilometers, would it make a difference if the enemy vessel is +Z or -Z axis? At those distances, the enemy would have to be thousands and thousands of kilometers above or below.
And in that case, the Enterprise would have to not only Z axis maneuver but likely change course heading (e.g. heading 122 mark 12), and this is just what Kirk did in combat.