I don't follow you here.
In this case, the motivation to restore the mine is power and the satisfaction of saving the world, at least this part of the world anyway. Whichever group takes over the mine will now grow in power all thanks to the heroes.
The status quo has changed for the better. This is a powerful motivating factor. It can provide just as much of a sense of accomplishment and heroism as preventing an ideal status quo from changing. It's even more satisfying, in my opinion as a player, because I had a more active say in the change.
Whoa there. So in your monster-infested mine example, the players' characters have pre-established motivations to gain power and provide assistance to would-be miners? In that case, the presence of the mine is - by your definitions and examples - a clear railroad, since it compels those players to take a specific course of action, or else go against their characters' motivations.
Or, if the characters do not have any motivations to deal with the mine, they'll just ignore it, and it becomes a valueless piece of fluff - a mark on a map, never referred to again.
That's the problem with your definition of railroading -
anything that engages with the PCs' motivation becomes a railroad.
Got a PC with a fear of drowning? Can't have any rivers or lakes in your sandbox, then - he'll feel constrained to go around them. Equally, they must be absent if a character has a liking for swimming, lest he feel constrained to investigate them.
Character afraid of heights, or a keen mountaineer? Your gameworld had best be flat as a pancake to avoid railroading him.
In the end, once you've eliminated anything which can possibly influence the PCs into actions contrary to the players' unstated wishes, your sandbox will be exactly that - just a featureless box of sand.
Unless any of the PCs have strong feelings regarding sand, of course...