Here's why there's no designated heir:
* The chalice is more of a tool than a possession: It serves as a vessel for the Silver Crystal's power, not a personal item that gets passed down.
* The Silver Crystal chooses its wielder: The Silver Crystal is a sentient entity that grants its power to those who are pure of heart and dedicated to protecting the world. Usagi Tsukino (Sailor Moon) is chosen by the Crystal, not because of her lineage, but because of her inherent goodness and potential.
* The chalice's power is tied to the Silver Crystal: The chalice's true significance lies in its ability to amplify and focus the Silver Crystal's power. Without the Crystal, the chalice is simply an empty vessel.
Therefore, the chalice doesn't have a specific heir. Its power is accessible to those who are deemed worthy by the Silver Crystal itself.