Fulltext searching on shared hosting - 2000 Entires with around 500 words each
I often query a 170K-row table using InnoDB FULLTEXT.  (Avg: 300 words.)  I typically get the answer of a few dozen rows in under 200ms.  That 200ms drops to below 20ms if the cache is warm.

I don't think the number of documents in the table matters as much as the number of documents returned.  If you need to fetch the text for 100 documents (after searching the index), that could take a full second to randomly fetch from a cheap disk.

Beware of benchmarks; they may be delivering only document ids, not the text; and they may be assuming everything is cached in RAM.  Since you have only a few MB, the caching is a valid assumption for you.