It's EXACTLY a railroad. Chapters are absolutely in mandatory order, and the 10 encounters of chapter 4 are exactly the same. And no matter what you do in one encounter, the next one will proceed exactly as planned. It's the perfect definition of ignoring everything that the adventurers do.
I think there are also a couple of definitions of railroad to use here, and only Chapter 2 doesn't qualify. Chapter 1 is a light kind of railroad, where there's a specific sequence of events that has to occur in order, and although there's nothing really stopping the PCs from trying alternate things to do, those things won't help the PCs progress further to their goals. When Volo tells the PCs to go to the tavern where he and Floon were getting drunk, the PCs could choose to do other things I suppose, but there aren't a lot of options they could come up with that would help them any better than following the obvious clue. Maybe finding a caster capable of casting Locate Creature and then combing the city for weeks trying to find Floon that way? Every encounter in Chapter 1 leads logically to the next encounter. This isn't the infuriating kind of railroading, mostly boring since the PC choices come up to "Follow obvious clue" or "Do Nothing". Granted, every published adventure is going to have this kind of railroading to SOME extent because not every single plot point needs to have diverging paths. "You're in a tavern, and a man in robes by the fire gazes in your general direction with an appraising look and gestures for you to approach".
So, Chapter 1 has a sequence of encounters that all breadcrumb obviously to the next encounter, culminating in a sewer dungeon that's mostly got a single path through it with almost all branches off that path leading to uninteresting dead-ends (hey, it's a metaphor!). Once again, this is not too horrible for the introduction, although it would have been nice to give the PCs some kind of interesting choices to make. Chapter 2 really opens up and allows the PCs to make some decisions. There are some complaints about it being a little too unstructured, and I think the running the tavern bit was heavily watered down and as written is a complete money sink for the PCs (which will never come into the campaign again as a money maker if you go straight to Dungeon of the Mad Mage after this).
And then Chapter 3 is back on the light rails like Chapter 1, except now it's a little worse. There's a sequence of events just like before, but the information flow isn't as obvious and not doing the right thing stalls the adventure completely. If the PCs don't investigate the explosion right outside their tavern or try to get the city watch involved, the adventure ends (still not too bad, because "PCs should be proactive" isn't too unreasonable an assumption). But, the next step involves the PCs investigating the explosion and deciding to visit the Temple that's associating with constructs of this kind. My players got here and wanted to track down the fleeing wounded guy who stole an item off the body and the Golem who threw the Fireball, but those aren't allowed options even though the event happened just literally 1-2 minutes before the PCs got to this point. Ok, PCs suddenly realize their only lead is going to the Temple that made these golems,whatever. And then they get to the Temple, and have to have a specific exchange of dialogue with the priest to get him to chastise the Nimblewright. And then the PCs have to, apropros of nothing, cast Detect Magic while the acolytes are hauling the Nimblewright's stuff out of the room. If they don't decide to cast the spell, or don't even
have the spell, the adventure ends. More realistically, the DM suddenly has to come up with an alternative solution to tracking down the Nimblewright on the spot. Now, the PCs have the ability to magically detect Nimblewrights, which isn't a terribly exciting sequence and has a good chance of sending them to the most pointless and boring roleplaying encounter I've ever seen solely so that the DM can chuckle to themselves about how their PCs don't realize it's actually a famous person they're talking to. Eventually, the PCs end up at the estate where the Nimblewright is staying and then see that a brawl is going down. Here, there are a few opportunities for the PCs to come up with some different options for how to approach the situation (even though the information is still very limited), but the end result is the same. No matter who the PCs help, the Nimblewright escapes (what happened to PCs being able to track it?) and the PCs have very little information than when they started as even the people they helped basically say "Thanks, now get lost".
And then you get to Chapter 4, which is the really bad kind of railroad we talked about where the PCs' choices will be overridden if they are
too clever. At the end of the Macguffin chase I guess you give PCs more choices now in how to obtain the three keys to get into the vault, but by that time my players were quite annoyed at the constant hurdle jumping and just wanted to get into the dang vault! Once you do, the vault is once again a dungeon with a single linear path with little in the way of interesting detours, followed by a conversation where the PCs are very limited in ways to resolve other than the one scripted path.
