Illuvial training is a way better response to rerolling
However I think instead it should be a Nintendogs/Neopets/Tamagotchi style game where you take care of Tier 1 Illuvials allowing them to train up particular stats.
(A basic mobile game that can keep players hooked)
-Food Can be crafted with the abundant plants found in the overworld allowing you to customize each illuvial with elemental and dietary needs being met.
-Training Uses energy that is gained by food to be able to train different stats individually by either playing a minigame or some training animation.
-Petting/Playing Allows the player to pet, and play with the illuvials giving each of them individual personalities to increase the lore and lovability of the Illuvials.
-Limitations I believe that you should never be able to train your way to maximum stat values, that weren't already innately given to the Illuvial.
(this allows maxed stated t3's to still be quite valuable since it needs to be completely caught and trained up at least 5 different times)