stiff garden
Before you go ahead and thumbs down this, in PokeMMO, which is basically a modded game that combines a bunch of the older games together to be an MMO, breeding is used to increase the IVs of their offspring, which burns the parents in the process. Additionally, there are items you can give each parent to lock an IV stat for a specific stat like attack, defense, HP, etc.
For example, if the male parent has a 31 IV in attack and the female parent has a 31 IV in HP, or any desirable IV, but a maxed IV stat is of course ideal, you could equip a specific item to each parent to guarantee that the offspring has both the 31 IV attack and 31 IV HP. The rest of the stats like defense, special attack, etc, is a random outcome between the two IVs for a particular stat. For example, if the male parent had a defense IV of 8 and the female parent had a defense IV of 22, an IV at random between 8 and 22 would be given to the offspring. Using these items in breeding isn't required, but they are typically used to create competitive and sometimes perfect Pokemon, but it gets very expensive to breed a perfect pokemon (in game currency of course, but I would imagine the economics of this translate closely to real money).
So for Illuvium, there would be a total of 6 craftable items that you could use to lock a stat for each or one of the parents (attack speed, health, movement speed etc.) There are plenty of videos on PokeMMO breeding if what am I saying doesn't make sense.
This would increase scarcity of illuvials by burning the parents and it would also increase the use of the illuvidex because players would be searching and sniping for specific illuvials with a desired stat to use for breeding.