Part of the problem with this kind of "RPS design" is that even when you can get the PCs to fit neatly into these categories, as you get to higher levels it applies to the monsters less and less. For example, what would you consider a Rakshasa to be - Fighter, Mage, or Thief? What about an Ancient Dragon? How about a Planetar? Or a Marut? Or a Death Knight?

And the PCs blur these lines too. Cleric was one of the first spanners in the Fighter/Mage/Thief works, but it's only gotten worse since then - Druids, Rangers, Bards/Red Mages, Monks, Artificers etc have all served to muddy the waters on the player side too. You have classes that can close the distance quickly and fight from range, or ranged classes that can defend themselves against chargers with ease. RPS just doesn't cut it for modern D&D, and the prior editions had even more convoluted tactical permutations.