I don't really know enough about this for a proper answer, but some parts necessary for FTS are features in Postgres. The inverted index is the base building block, and Postgres does have this as GIN (generalized inverted index). The Postgres version is useful for more than just FTS, but as far as I understand it's pretty much the same thing under the hood as what dedicated FTS engines use.

From what I've read, the biggest weaknesses in Postgres FTS are in the language-specific stuff and not the core indexing. And especially in ranking results, where there are much more advanced methods than what Postgres uses.