I spent most of last week looking at live chat tools and also came to the same conclusion re ChatWoot (ie: good product + open source + lacking some features + but possibly extensible)
We're moving off of Intercom. Great set of features - but the pricing is horrendous.
I think live chat is more important than just the live chat. Using the chat widget for analytics, tracking, and automated messaging is also important.
I want to be able to take screenshots of what my users are looking at when they need support. I want to be able to detect what pages theyre on and provide contextual help when they need it.