In the web3 space, everyone is looking for opportunities to make money, so maintaining a healthy economy is a key factor to this.
Currently, illuvials caught are only useful in arena mode which may not be successful enough to have demand for purchasing illuvials.
Gauntlet mode could be part of the fix if tournaments/some other ladder required the use of self-owned illuvials. This would make it so players actually need to purchase harder to obtain illuvials for their teams. However, this only makes the higher tier/stage illuvials more valuable.
There can be an additional system where a player can re-roll the stats of an illuvial by merging x amount of the same illuvial. That player could also have a chance to lock in one or more stats of the three current illuvials by using a new material. This material can only be obtained by burning illuvials. Higher tier illuvial = higher tier material. higher stage = more material.
As it currently stands, I don't think there's a big enough sink to sustain a proper economy for illuvials. Even with the above system, it'd highly depend on how successful/popular gauntlet mode is. If people don't want to play it, then illuvials won't get purchased.