I think what needs to be accepted is that railroading can be fun. It can work.
Absolutely.
A published scenario is a railroad.
Not necessarily. There are lots of published scenarios that aren't railroads.
Masks of Nyarlathotep,
Keep on the Borderlands,
Isle of Dread, and the
Banewarrens all leap to mind (with varying degrees of non-linearity).
First, we assume the PCs actually WANT to go to the caves, because quite easily here the DM could use the power of the choo-choo to force them there. (...) Have we hit our first railroad: the assumption that since the adventure is based around the Caves of Chaos, we are actually GOING to the caves of chaos?
The ability to choose which opportunities to pursue is pretty much the definition of a sandbox campaign. So, yes, if the DM predetermines what scenario the PCs are going to engage with each week then you are not, in fact, running a sandbox campaign.
With that being said, most people consider enforced scenario selection to be the lightest form of railroading and pretty much inoffensive (assuming you didn't promise a sandbox campaign).
Pretty much everything else you described isn't a railroad (although it's not really a sandbox, either), with one possible exception.
I mean, the DM has an end-point in mind (stop the cultists before Armageddon) and the PCs are going to follow the bread-crumbs to get to it. At any point, they can deviate from the trail to do other things (slay a dragon, start a keep, marry the princess, etc) and possibly even give up altogether (and face the consequences of an evil army marching on the realms of man).
The "Armageddon scenario" is kind of a tricky one to judge.
If I said: "I totally gave them a choice: They either do what I want them to do or their PC will definitely die." Then it's not particularly difficult to see that I'm not really offering any sort of legitimate choice.
The Armageddon scenario similarly says: "I totally gave them a choice: They either do what I want them to do or I blow up the entire world with them in it." It seems like a similar dishonest is being manifested in this so-called "choice".
OTOH, I don't think "I'm threatening something you care about" automatically equates to "I'm railroading you".
There's a legitimate grey area here. But if you sat me down and forced me to pass judgment on some hypothetical example the first thing I'd look at is the specificity of the action being "forced". There's a scale between
(1) "you've found the One Ring, Sauron is seeking it"
(2) "you've found the One Ring and Frodo, specifically, must carry it to Mt. Doom"
(3) "you've found the One Ring, Frodo must carry it, and he must go to Bree, Rivendell, Moria, and Lothlorien in that order"
With less (or no) railroading at one end and a lot of railroading at the other.
What I will say is that Armageddon scenarios tend to preclude sandbox play specifically because the priority
demanded by the Armageddon scenario tends to preclude freedom in scenario selection. (Although this doesn't necessarily have to be true: For example, World War II can be going on in the background without the PCs feeling as if they're personally responsible for stopping Hitler.)
They may want to go south and explore swamps, check out Quasequenton (which the module strictly tells the DM to railroad the PCs away from if he's not prepped for it)
Huh? I don't remember that in B2 nor can I seem to find any reference to it now that I'm reviewing the module. Is this something from
Return to the Keep on the Borderlands perhaps?
Final note: I think part of your confusion is probably stemming from treating "sandbox" as being the polar opposite of "railroading". This isn't particularly true. The defining trait of a sandbox campaign is freedom in scenario selection; but, as I noted above, this is generally considered the
lightest form of railroading, not the most severe form.
The other source of your confusion may lie in trying to determine whether something is a "railroad" by trying to identify whether it has a "plot"; and then specifically trying to identify the "plot" by describing the sequence of events as it occurred at the game table. The problem here is that everything has a plot
after the fact. Railroading happens when the GM attempts to enforce a sequence of events which has been
pre-plotted.