I would personally love this as a site admin as well. It would be a great way to consolidate user interactions—rather than have chats in the community and separate chats as part of a customer support solution such as Intercom, have everything in one place.
As an end-user, I would say it depends on the interaction. If I am having a small interaction with a website that I expect to be a one-time conversation, I wouldn’t really care much about what technology is backing that interaction. However, if I am interacting with an entity with which I communicate regularly or in which I am genuinely interested, it would be wonderful if that initial chat interaction led to a community which I could explore and join. Perhaps chats could then turn into PMs or topics for longer-term communication.