I have a beef against some so called 'sandbox' DMs.
Please don't take this the wrong way, but sometimes it seems you have a beef with a lot of things.
There are some DMs out there that think they run a sandbox, but they don't. The secret to running a sandbox is that you have to be willing to prep vastly more material than you need or intend to use. Indeed, that is my definition of a sandbox. If you aren't prepping material you don't intend to use, you aren't running a sandbox. In a true sandbox, wherever you go, there is the fun, and eventually players get enough experience in the setting that they can orientate themselves within it and start being proactive and deciding what goals that they have, what factions that they intend to ally with, who they think the villains are, and so forth. One conceivable way to run a sandbox for example might be to have a stack of low level modules, and you've drawn up a setting that scatters these low level modules liberally all over a fairly small homebrew map. You might need to write 16 different adventures each with plot points that might lead to further adventures. The point is that sandboxes are informationally dense. They are not for the faint-hearted DM, but something to undertake once you are experienced and if you are the sort of person that world-builds continually in your leisure time because it's one of the most enjoyable things you do.
To some extent I disagree.
What needs to be fully prepped, at least to a solid framework level, is the setting: the cultures, the maps, the history, the pantheons, some key people both past and present, and so on. Having a stack of adventures handy, be they canned modules or homebrew ideas, is also useful; but other than vague ideas I don't think these all need to be pre-placed.
My current campaign started this way. I had nearly all the framework in place before puck drop, and had a couple of players who already knew how they were going to cause a party to form around their PCs - a Bard and a Cavalier rolling up-country from village to village, with at each stop the Bard singing about the heroic deeds they had (not in fact) done and the heroic deeds they were going to do "up in the mountains" while the Cavalier tried his best to look impressive and heroic, and asking for brave and sturdy adventurers to join them - and at about one PC per village the party formed. Up in the mountains awaited B2 Keep on the Borderlands, one of a very few adventures I had hard-placed but only after hearing what these players had in mind.
Once 'Keep' was done they went back down-country and split the party in two (i.e. I started running two different groups of PCs with some overlapping players). One group proactively looked for something to do (and found it, getting into a variant on A-series) while I had to put something in the path of the other group. Since then it's been a mix of "let's go this way and see what's there" and "here's an adventure hook or ten"; and ten-plus years later it's still going.
But in my experience a good half of the people who run a sandbox do so because they think its a way to cut down on their prep time.
During play, it is; because you've done all the heavy lifting before the puck ever hit the ice.
Instead of seeing it as a style where they will need to produce vastly more content than they'll need, they see it as a style where they don't really have to prep anything because they can improvise it on the fly.
Exactly. Ideally, if the setting prep is solid enough the macro-scale game largely runs itself once it starts, leaving me to worry about the run-of-play minutae such as statting up monsters and keeping notes and logs.
And invariably, they are suffering from Dunning-Kruger effect and are vastly less capable at improvising than they think they are, because the secret to good improvisation is that you've prepped so much material that its easy to adapt things you've already done to the moment. For example, further up the thread someone suggested that the material in an adventure path became useless if people got off the path. And that may be true if your dealing with a less experienced DM, but it's perfectly easy to take adventure path material and plug it into a sandbox. In the case of the best as I can tell so far, completely on rails "Skull & Shackles" campaign, it wouldn't at all be hard to pick up a couple of relevant Golarion sourcebooks, and if the party got off the rails and went sandbox on you, you could accommodate that. The material that you didn't use from the Adventure Path, could still come back at a later point, just with a different timing and setup. Instead of A->B->C, you might do A->B->H->X->C->K->D. The veteran DM would be able to flex.
No idea what Dunning-Kruger effect is (don't bother explaining it

) but in general I agree with the rest of this.
In the hands of an experienced DM, modules become setting guides.
Maybe, but I'd go further and say that an experienced DM will already have the setting in place and be easily able to slot the module (or AP) into that setting such that it makes sense for it to be there. (for these purposes let's ignore modules or adventures such as Q1 Demonweb Pits that intentionally take the PCs off-world or somewhere equally remote)
All DMing involves a certain amount of illusionism, and the trick is to be able to pull off the illusion so that to the players the world is real and has always been there and is living and changing and responds to what their characters do. The key to success as a DM is to give the players the experience of exploring something, and discovering secrets you have been hiding on your side of the screen.
Exactly.
In my experience, no improvisational DM is able to sustain the illusion. So many tricks that improvisational players tell you to use, destroy that sense you are exploring a real world. Things like taking your queues from the players own expectations, quickly cause the players to realize that nothing is real. The joy of exploration is not finding what you wanted to find, but discovering what you didn't know was there.
I agree, to a point.
Sometimes it works out well, however, if you take a player's cue (not queue, that's a lineup

) or idea and stow it away for long enough that the player doesn't remember she gave it to you, and then use it. Example: a player in my game had an idea for a variant monster - a gelatinous cube that hung from and moved along the ceiling, to avoid the clean-floor hint that a cube might be about. I took this little piece of brilliance and stowed it away. Half a real-world year later a party is in a dungeon and notices there's a "tide-line" about six inches off the floor, above which the walls are clean and below which is lots of dust and dirt. They figure (on only that evidence) it has something to do with periodic flooding and carry on - and guess whose PC walked face-first into the cube? Yep, the same guy who had suggested it in the first place! (even better was that I had to tell him after that it was his idea, as he'd completely forgotten ever suggesting it)
Guys that run games on random encounters or other random generators, never produce anything as interesting as the GMs that use those same random generators before they start play to inspire and produce content that then flows off their tongue when called for. If your random encounter table isn't imbedded in an already heavily detailed world, then it is never going to produce anything particularly rich.
I wouldn't say never, but I would say the result would tend to be - surprise surprise - more random.
I for the life of me don't understand why anyone would think that a group of players thrust in to a setting which even the DM does not clearly see would have enough information about the setting to make good choices about what they want to do. If you are running a campaign, just as if you are writing a novel, the first page is the most important page of the book. I don't care how clever your setting is, if that cleverness requires overturning a random rock on hex grid A10318 because it isn't believable that the PCs would find something that's been hidden for 1000 years, you have a completely wrong-headed idea of what running a game is like, and it's not your players fault that they didn't go looking. Paint the damn scene. Put up signposts everywhere. Have the adventure come to them. Then, once they've become locals and know the lay of the land, they can make choices about where to explore.
The fault there lies in having whatever's under that random rock in hex A10318 be so important to the plot that it has to be found. A true sandbox might well have something like this, but the continuation of the game/story/plot wouldn't rest on it being found.
Do you have an obligation to have the adventure come to the players, even in (especially in!) a sandbox? Yes. A thousand times yes. And anyone that tells you otherwise isn't worth listening to. Shove the adventure in their faces, and keep shoving adventures in their face until they take a hook.
This is one thing.
Do not expect the players to find your content without a map. Do not expect the players to understand your setting without a detailed guide. Do not expect the players to care about the setting until after they are immersed in it. And above all, do not expect your players to create your content for you.
This is another.
If all the second part there is done right the first part shouldn't be as necessary, as there'll be enough of a setting - including legends and lore - in place to pique some interest. The difference is that interest will probably go in an unforeseen direction, thus keeping the DM on her toes.