It should be possible to search the contents of all documents.
To allow searching the documents contents efficiently, a separate index is necessary. The "defacto standard" for fulltext search on the JVM is something backed by Lucene. Another option is to use a RDBMS that supports fulltext search.
This adds another component to the mix, which increases the complexity of the setup and the software. Since docspell works great without this feature, it shouldn't have a huge impact on the application, i.e. if the fulltext search component is down or broken, docspell should still work (just the fulltext search is then not working).
If docspell is running on PostgreSQL, it would be nice to also use it for fulltext search to save the cost of running another component. But I don't want to lock the database to PostgreSQL only because of the fulltext search feature.
ElasticSearch and Apache SOLR are quite similiar in features. SOLR is part of Lucene and therefore lives in the Apache ecosystem. I would choose SOLR over ElasticSearch, because I used it before.
The last option (supporting all) is interesting, since it would enable to use PostgreSQL for fulltext search for those that use PostgreSQL as the database for docspell.
In a first step, identify what docspell needs from a fulltext search component and create this interface and an implementation for Apache SOLR. This enables all users to use the fulltext search feature. As a later step, an implementation based on PostgreSQL and/or ElasticSearch could be provided, too.