Hi, I started an Elasticsearch hosting company, since sold, and have built products on PG's search and SQLite FTS search.
There are in my mind two reasons to not use PG's search.
1. Elasticsearch allows you to build sophisticated linguistic and feature scoring pipelines to optimize your search quality. This is not a typical use case in PG.
2. Your primary database is usually your scaling bottleneck even without adding a relatively expensive search workload into the mix. A full-text search tends to be around as expensive as a 5% table scan of the related table. Most DBAs don't like large scan workloads.
Do you see any particular reasons to use or not use ZomboDB [1]? It claims to lets you use ElasticSearch from PG seamlessly e.g. it manages coherency of which results ought to be returned according to the current transaction. (I've never quite ended up needing to use ES but it's always seemed to me I'd be likely to need ZomboDB if I did.)