Let's discuss the Copperajah in the room: the concept of a "rival" simply does not work when the character they're supposed to be a rival to is a silent protagonist. Like any other character dynamic, rivals need personalities that play off of each other. Look at Goku and Vegeta, or for a Pokemon-related example, Ash and his various rivals in the anime. Those work because both characters involved have their own personalities and thus the rivalry isn't entirely one-sided.

In Pokemon games though, your "rival" is the same as any other random trainer you fight, the only difference is the amount of times you fight them. Any actual "rivalry" is entirely in the player's imagination.