I'm not a percussion player. Hopefully you get an answer from an experienced percussionist or band director.
My first thought was along the lines of Andy Bonner's comment: the snare drum may get the most important parts and so responsibility for just the single drum might help the player's focus on their part.
More specifically, I think the ensemble treatment is different when comparing a trap kit with percussion band. At the risk of oversimplifying, it isn't that hard to coordinate hitting the snare on the back beat for a rock beat on a trap kit. When fills are played, even moving all over the kit, the snare tends to drop out or strokes alternate between snare and the other drums.
But in a percussion ensemble it's often different. There can be really active snare parts, where the strokes are alternating on the one snare. If that snare player had to simultaneously play accents, counterrhythms, etc. on other drums, it becomes too much for one player.
Basically, it's two different kinds of playing for the snare. The drum kit has a lot of drums, but it's not really used the way percussion ensembles use multiple drums/drummers.
Compare a trap kit beat...
...to an ensemble...
It's not that the first one, the trap kit beat is easy, but you might say it's economical. It gets a lot done with only a little bit of unison playing. One person can do it. But the second one, there's just too much unison stuff. (I think unison is the correct term in percussion, when one or more hits happen simultaneously.) One person can't play that ensemble piece. The snare player would not be able to take on additionally even one of the other parts.
Following @Edward's suggestion I found a transcription of some Blue Devils music...
...like I said, I'm not a percussionist so I can't really say about the difficulty, but there are indications for strokes. That transcription is from a YouTube video.
Apropos of nothing, I really wish the NFL had drum lines or marching bands playing during breaks.