This reminds me of a character I half-created specifically to justify always being in interesting scenes, but also just as a way of always being of use to a fellow PC (or to an NPC): the cosmic butler.
He's just so supernally good at butlery that little things like time, space, and the laws of physics won't get in the way of him being on hand with exactly the items and services his master requires or desires precisely when they're wanted.
The cosmic butler hands you the exact drink you want the moment you start to consider wanting it, and it's what you wanted even if you weren't sure what you wanted. He is there, holding the door as you approach, and just has pulled the car up and is holding THAT door by the time you walk down the walkway to it. The ultimate batman, he has your clothes laid out and ready for exactly the occasion you need. He can offer you exactly the golf club or hunting rifle or high explosive grenade launcher you want at exactly the right time; loaded, oiled, and ready to use.
The cosmic butler has impeccable small talk, advice on all the important people in any event or gathering, and is never ruffled. He is always just finishing any menial task necessary just as you happen to enter a room, and looks dignified while doing it.
And, if asked how he did it, his usual reply is something along the lines of, "I couldn't allow [insert crippling impossibility here] to inconvenience you, Sir. It would be unprofessional."
I quite agree, especially with that last sentence.
Other posters have addressed this, but I'll throw in my two cents as well.
Plots are not railroads. Railroads aren't even always plots (but more railroads are plots than plots are railroads).
Railroads are linear; the PCs have no (or very limited) agency, following a prescribed path from beginning to end. If they have choices, they're "pick which branch of this path you'll take" choices, not "how do you solve this problem?" choices.
The DM preparing encounters doesn't make it a railroad.
A railroad requires that the PCs be only allowed to go to this encounter first, that they must resolve that encounter in a particular way to get to the next encounter, which they must face in a particular way to get to the next, etc. etc.
What you've described here as a "railroad" is not one. It's a sandbox, assuming the DM has prepared his various encounters and NPCs such that they'll react organically to the PCs' choices.
The question to ask yourself is, "How many choices do the PCs have, and how meaningful are they?" A railroad will have only 1 meaningful "choice:" Pursue this in a precise way. A branching railroad will have several meaningful choices, but they'll be "multiple choice," and you can't pick anything off-menu. Picking a choice just sends you down one of several branches. (Most video game RPGs which offer choices to the PCs use this technique, and have spiderwebs of myriad choices, some of which re-cross and some of which branch off forever. And some RPGs have a pseudo-sandbox of mini-railroads, which is what you're thinking of, most likely, when you say that the railroad starts when a plot is picked.)
A sandbox is a "fill in the blank" question as opposed to a railroad's "multiple choice" question. (A false sandbox will be a railroad where the "blank" only has one right answer, and the players must guess it. A true sandbox has no 'wrong' answer, in that the world will react organically to anything they do, and the 'rightness' of the answer only varies based on how well it leads the PCs towards their goals.)
So I think the fundamental problem is that you, Darth_Ultron, have a false definition of both "sandbox" and "railroad."
Which is what I keep trying to make a point of. Every time you characterize a sandbox as "random," you're missing the point of the term entirely. And, moreover, you're mischaracterizing railroads when you try to say that any preparation at all leads to one.
It's only a railroad if the PCs' choices are planned for them. It's a sandbox if the PCs can do anything, and the encounters are prepared such that they can react to whatever the PCs are doing.