That space gets filled up a lot more with the tree of variations, when one shows up. But anyway, yes, that is broadly true, the goal of this layout is not exclusively to avoid empty spaces, but to make chat a lot more usable without making other component any less usable.
Yes, I think that would look better: board in the middle, chat on the right, everything else on the left. Unfortunately it does not seem like that arrangement is possible by only manipulating the CSS, or even the HTML structure. It would probably require a more indepth change on the OGS frontend.