Having everything bagged and organized accordingly should prevent any messes (for most boxes at least). Even for board games, the pros for storing vertically outweigh any that storing horizontally would have, like ease of access and less wear and tear, for example. Gravity doesn't work particularly well when stacking and piling, especially for big collections, or for any other type of item, really.
Just my two cents, of course, we're all a world apart after all; but I can see now what you meant when you said that "a board game collection will never look anything but chaotic".