The kubernetes StatefulSets project is actually trying to build a usecase for Postgresql-on-s3. I think its the wrong abstraction to go after in most usecases for database systems including things like ElasticSearch. I dont think that even Aurora is trying to solve this using persistent storage.
You dont really care about solving underlying persistence... you make the whole system fault tolerant : which includes high availability, failover, etc.
IMHO kubernetes still has problems in orchestrating an ElasticSearch/Zookeeper cluster for high availability. The problem of discovery, load balancing, etc are bottlenecks.