There's a fourth possibility as well:
(d) the disease is merely serving as a doom-clock to keep the party moving, perhaps more quickly than they otherwise might, and is a side piece to the actual adventure. For example they think they're going out after the cure and their odds are very good of finding it, but the real adventure will reveal itself during this process e.g. finding the cure draws them in to Castle Amber or Ravenloft or somewhere else they then have to complete an adventure to get out of.
This might relate to your point c above, but it could also just be a means of (in theory) preventing the 5-minute workday approach.
I would be....more than a little suspicious of using it as a "death clock." Yes, that has the legitimate purpose of ensuring that the party keeps moving. But that could also
extremely easily be just a clever way of disguising pretty blatant railroading.
As for the Castle Amber thing, that's just using one hook (cure search) to trigger another (Castle Amber). Whether that is crappy "you get no choices but I will deceive you into
thinking you have choices" behavior, or perfectly acceptable "alright, we've closed one leg of this journey, a new one now begins" is wholly dependent on how it plays out. As noted, it's not really railroading if genuinely nobody ever desires nor attempts to depart the rails. Maybe that's because they tacitly know and are on board for this short trek of railing before getting off to explore, maybe it's because they're just too caught up in the moment, who knows. Any reason is fine, the players have still exercised their choice to embrace the game on offer.
Railroading occurs when you pretend that players can exercise choice, while actually denying them choice. That's it.
Context matters, and so does frequency: is this disease business a one-off for this adventure only (in which case it's probably OK no matter what) or is it, or something similar, a feature of almost every adventure in the campaign (in which case it's very likely bad)?
I very much disagree that a single use is nigh guaranteed to be legitimate. That is simply not true. There's a quote from Garak of Deep Space Nine fame which is very relevant here. He (a Cardassian) listens to the story of the boy who cried wolf, and Dr. Bashir concludes with:
B: "The
point is, if you lie all the time, no one will believe you."
G: "Are you sure that's the point, Doctor?"
B: "Yes. What else could it be?"
G: "That you should never tell the same lie twice."
Instead, I would say that the way that it is used tells us a lot. Because the tool itself is neither good nor bad... it's just very, very easy to use it for a bad reason. Like a tool of violence (assault rifle, IRL longsword, etc.) where it has little to no utility function but extremely high violence potential. It's not that these things are inherently bad, they aren't and can't be. But they have few legitimate uses and many, severe, bad ones.
Hence, if the players get why you're doing this and (tacitly) agree it's legitimate, or if they're so swept up in the moment that it never occurs to them to do anything else, then cool. They're clearly on board this train, they aren't being
railroaded, they're being
transported to where they want to go.
It's when the players don't actually agree and aren't actually swept up in it...but the DM uses tools like this to
coerce their agreement out of them, no matter what...that we have a problem. And that can absolutely happen even if the DM only uses any particular tool once and only once per campaign.