Correct, that would take more than 5 minutes, although still possible to do with Faiss (and not that hard relatively speaking - in the Teclis demo, I indeed did your second point - combine results with a keyword search engine and there are many simple solutions you can use out there like Meilisearch, Sonic etc.e). If you were to try using an external API for vector search, you would still need to build keyword based search separately (and then combining/ranking logic) so then you may be better off just building the entire stack anyway.
Anyway, for me, the number one priority was latency and it is hard to beat on-premise search for that.
Even then, a vector search API is just one component you will need in your stack. You need to pick the right model, create vectors (GPU intensive), then possibly combine search results with keyword based search (say BM25) to improve accuracy etc. I am still waiting to see an end-to-end API doing all this.