There was an interesting related talk on SIGMOD this year[1], at which Microsoft presented how they take db-level index suggestions from many MS SQL databases in Azure, apply most of them to the databases on hidden replicas, do A/B testing on them (replaying recorded workloads on replicas, etc), and push only the proven suggestions to production.
Seems like tools described in the article could allow doing something similar with PostgreSQL at scale.
[1] https://www.microsoft.com/en-us/research/publication/automat...
I was recently looking into this exact topic and found Dexter[1], which also uses HypoPG.