While I'm not sure how much I like it, it is possible to put the chatbox in a side panel. See this forum uses SMF and chatbox on the side:
https://www.tinyportal.net/index.php?action=forum. They have a quite different theme than we use (and possibly a different chat mod) but you can at least see how it looks.
But anyway, if others would like, we certainly can try it. Just because I'm old and I don't like change doesn't mean we can't change things. I just like to see more people wanting the change than don't want it.
Oh, another thing we could do, is put the discussion from the chat on top of the box where you enter text, more like IRC works. I like it this way, because you read downward, to follow the discussion, just like we normally read English. For IRC, it bugs me to have to read upwards, to follow the discussion. (!idea - I wonder if my irc client allows to switch it around....?)
Actually, I think I can switch the position of the discussion vs the text field, and change the direction of the discussion, separately. I think we could have both worlds. But again, only if more want it than don't, or if it would prevent confusing it with search. I'm not sure if it would help with confusion.