I run in a very improvisational style and yet ran Avernus which is railroady as all get out*. Improvisation meant weirdo NPCs and random encounters, but campaign stayed on the rails.I don't know if the distinction between improvisational and planned games fits into this discussion. While I think most enjoyable improvised games would be sandbox, you could also have one that follows another parameter - but the distinction of whether a DM has prepared material or is off the cuff doesn't impact how the story is laid out... just when it gets laid out.
However, I think that a game that has the DM respond to the players choices would generally be sandbox.
Are you seeing it differently?
*I mean, the flowcharts in the modules are branchless lines!