You seem to be saying the alternatives are "sandbox" and "get on the plot train." I disagree with that. There seems to me to be more middle ground than I get from what you say.
These are two extremities on a scale, But note that I've always been saying that it's very rare that games are at one extremity, for me it's always shades of gray along the axis.
What I absolutely refuse is people badwrongfunning people playing towards one end of the scale, namely the more railroading part of it.
For example, I've ran ton of games for kids and beginners with any edition, these were in no way inferior to anyone else's game as everyone had great fun with the game. But these were decidedly more on the railroading side of the scale.
The story belongs to the PCs. Maybe you present them with several possible things to do, and maybe some of those are from one backstory or another. They'll still choose what appeals to the party overall, and that's fine. I haven't found "This is (for example) Taman's nemesis" to skew intra-party balance in any meaningful way, so long as it wasn't for the whole campaign. Experiences and tastes vary, of course.
And yet, in at least two major campaigns that I ran (I'm saying campaigns running for scores of sessions over more than one year), the backgrounds and the resulting intrigues became preponderent, which in a sense is good as it gave player "centricity" but every time we focussed on one player's story, it was at the detriment of all the other players stories. And these were nice people, playing fair, and wanting to share the fun. But in the end, our common agreement was that it was in general better to run a central DM-led story with large contributions from players with side stories.
Note that this is a topic that we studied in length in our LARP association, when you have 250 people playing, you need to control the stories, because we had some cases in which players either invented something completely and derailed (sometimes on purpose) other teams, sometimes for hours (when you are playing over 10 km^2, with that many people spread all around the place, it's hard to control everything), or cases where some players just started following another team's leads and spoiling/destroying the quest.
So we needed ways to make sure that the stories were used properly, and invented a number of structures, the pyramid, the pierced pipes, the double torus, etc.
While too much is too much, I want the players to put stuff in their characters' backstories I can pull from, to tie them to the setting and the campaign--to make the latter ahout the PCs.
Note that is a somewhat different matter, a perfect railroad can totally be completely about the PCs...
I have never seen a player railroad a D&D (or D&D-alike) campaign. Not one time. Every published adventure I have taken part in has been (or felt like) a railroad. Every GM-authored "uber-arc" in any game I've been in has been (or felt like) a railroad.
Collaborative experience. Shared responsibility. My gig is to bring a world and scenarios.
So you do have scenarios. So if they follow such a scenario, are they being railroaded ?