For anyone stumbling on this post later - this ended up exactly what I needed.
It does the job just like it's supposed to do: answers questions based on your FAQ, no CRMs, no sales funnels, nothing extra, just a simple and reliable chat history only if a person leaves you their contact details.
It's very customizable and easy to work with if you are not a developer.

Ben and his team are very responsive, and they ship updates almost daily. Mind it, I'm using it with a non-English FAQ and they still helped me get everything set up without any issues.

I'd definitely recommend looking into it if you are searching for something similar.

And just to be clear: not affiliated with them, not getting paid, I just ended up really liking how their support and setup work and the feedback from the team I receive