The special abilities that came with stats reaching a certain point were hardly used in high-level battles, almost to the point where they seemed forgotten about.
However, that was not the case. The issue here was expectation versus reality.
As was shown through Ash's battle, when two parties had the same special ability - or around the same - activating it was pointless because it would not give either party any real benefit. It would just drain resources and exacerbate the stalemate.
As such, it meant that one could only utilize it in battles against someone stronger or someone weaker. However, another problem cropped up here.
Against someone stronger, it would be useless as the person would then activate their special ability which was one stage higher than yours, increasing their already sufficient advantage.
Against someone weaker, you were already dominating them, so was there really a need to use it?