We have a rather difficult onboarding process and users often message via the chat for help.
For the homepage I'd say visitors message rarely so it is less useful. That said, the ones that do are usually the same who convert as they are already fairly qualified leads and just want a little extra info before they sign up.
Back when I was doing web stuff for clients I got a lot of help through hostgator chat function and it was great. It all depends on how knowledgeable the person on the other side is. The medium is fine in and of itself.
One - the person on the other end works for a different company and they can answer a few common questions, but everything else is "call this 800 number." Cell phone companies do this.
Two - the person immediately says "give me your phone number and lets talk on the phone" (car dealers are terrible for this).
I guess there is a third type - companies using a laughably terrible bot. I encountered this with Sony after I bought a game online and it wouldn't start. I eventually called in and they instantly refunded my money because I think it was a common problem.
There is a fourth type. The MVNO I have my phone plan (and friends/family) with actually utilizes their live chat for support, and usually I get a very knowledgeable person in 30ish seconds. It's great. Red Pocket Mobile is the name.