Hi Stephen,
It's an interesting idea. As you might have noticed, the chat doesn't have a separate page, so it's not just about adding a link somewhere. What current chat icon does is it uses Javascript to display a chat window div-element.
I think it is a good improvement to be added to the system. I will put a ticket for it for us to have a look at at some point and will keep you informed if we have updates on it.
Yuliya