It's really hard to coordinate between LLMs, a vector store, chunking the embeddings, turning user's chats into query embeddings - and other queries - etc. It's a complicated search relevance problem that's extremely multifaceted, use case, and domain specific. Just doing search from a search bar well without all this complexity is hard enough. And vector search is just one data store you might use alongside others (alongside keyword search, SQL, whatever else).
I say this to get across Postgres is actually uniquely situated to bring a multifaceted approach that's not just about vector storage / retrieval.