I would say that it is obvious that fuller campaign interaction is part of the desire to play in a sandbox game, but IMHO you have simply failed to go far enough with your analysis.
The exclusion of sandboxing was, indeed, deliberate. It wasn't because I don't enjoy sandbox play, for certainly I do, but rather I didn't want to automatically install sandboxes as the highest form of D&D without examination of the levels of engagement with the campaign I discussed in my original post.
The situation is somewhat complicated - at least to me - by an unclear definition of what a sandbox is. Some areas that I might argue are inside or outside the definition of a sandbox are disputed by others. So, no sandbox as the next level - at least in the OP!
One aspect of the campaign that fascinates me is from where comes the motivation for the adventures the PCs come on: Is it imposed from above from the DM - such cases of running APs, GDQ, I3-5, this is a motivation imposed on the PCs. Conversely, when you run a setting where you've described the home base and what lies around... but then leave the choice up to the players. To then go even more in the direction of player motivation... when the players actually come up with the adventure ideas (we want to explore an orc lair in the mountains; there's one there because we say so!)
There is no doubt that some form of adventure structure as imposed by the DM is popular; see the GDQ and other adventures of its ilk. (Personally, I'm even more fond of Necropolis). These adventures read as connected sandboxes: you have a great deal of freedom within each sandbox, but limited freedom in your progression along the chain. There is an overall linear structure to the entire beast, albeit not as restrictive as many APs.
A related question that poses itself is this: in campaigns where the group is always travelling and not returning to the same locations (see especially GDQ, but others likewise), how much engagement and devotion can the players gain towards any one location? There is much to be said for the settled home base: when you spend a lot of time in one location, it can become a lot more real than the "drive-by" aspects of places like the Shrine of the Kuo-toa.
(One of the great aspects of the Vault of the Drow is how much time you could conceivably spend there, although I wonder how many DMs and players were good enough at the game to run it in such a manner; sadly, I've never run GDQ, although I've played or run most of the other great classic series).
I know that in my play of the Savage Tide AP, we really got a great relationship with the home town Sassarinne going - but, alas, such did not reoccur when we reached the Isle of Dread and had the new settlement. This is in stark contrast to my Great Kingdom campaign where the group made three trips to the Isle of Dread and had much rapport with the isle; interestingly, it was based on the creatures and situations to be found there, rather than NPCs, so a pleasing level of engagement with the game can be found even without PC-NPC interaction; occasionally the situation is enough!
Cheers!