I think this is a relatively key crux: If you look at a description of player-directed action and see nothing but chaos, then a sandbox campaign isn't going to make sense to you. If you feel that the only way for a campaign to have structure or form or purpose is for the GM to supply it, then a sandbox campaign is essentially incomprehensible to you.
Why is it that sandbox proponents just
love to put the words in other people's mouths?
Sandbox campaigns
do not have a structure. That's the very definition of the sandbox - go anywhere, do anything, without any constrains except those set up by the world itself. Structure is a function of the story. A sandbox game may end up with a loose, somewhat coherent story, but that's a side-effect, not a purpose unto itself.
The players themselves are like a herd of cats. Left to their own devices, they'll pursue their individual agendas, which may sometimes coincide for long enough for them to work on a "common goal." But the game described in the essay which I was referring to in my post has a multitude of players, playing a multitude of characters, with
no structure whatsoever, because each session is scheduled by the players, who also decide which characters to run, where to play, and, pretty much, what to do. The DM is there just to provide the scenery and what simply cannot amount to a very complex adventure, since it looks like each adventure must be concluded during the session, otherwise the whole freeform thing doesn't work. Maybe I'm wrong and people who managed to read more (I admit I'm not one of them) can explain in a couple of sentences how that works.
However, while I am most definitely disputing the ability of sandbox games to possess structure, I'm not complaining about form or purpose (both of which are very clear). I understand it completely, I just think it requires an extraordinary set of players and a highly unusual (and incredibly competent) GM to work.
For the record, I've played in sandbox campaigns. I loved the freedom, but every single one of them (bar one) devolved into terrible chaos and the GM finally had to terminate the campaign because individual player agendas became too much to handle and groups fell apart. I've also used sandbox settings (such as Wilderlands of High Fantasy) to complement my own games (I described my GM style in another post, if you need to look it up).
Incidentally, I'm sure that even a bad GM can run a sandbox game which revolves around players visiting random dungeons and looting treasure - which most "old skool" AD&D adventures were. I'm simply not interested in that style of play.