Postgresql is nowhere, NOWHERE near as powerful or as fast, but it'll work for the cases where you're doing simple search-interface bolt-ons to an existing product already using PGSQL.
Postgresql fulltext and ElasticSearch are ultimately solving two very different sorts of problems.
The only case I can think of where you would prefer to use PGSQL is if you some weird blending of normal relational data with the documents, but I think realistically you can sidestep that. We've been able to do pretty complicated queries against our ElasticSearch instances.
As you say they solve two very different sorts of problems.
If you want to do full text searching of documents, probably PostgreSQL is not the right answer. If you want to pull all customers in the UK with notes searched full-text-wise for "did not pay" then PostgreSQL is much better just because it all integrates into the same filter.
Postgresql fulltext and ElasticSearch are ultimately solving two very different sorts of problems.
The only case I can think of where you would prefer to use PGSQL is if you some weird blending of normal relational data with the documents, but I think realistically you can sidestep that. We've been able to do pretty complicated queries against our ElasticSearch instances.