Well, the reason it's good for data transfer is the same reason it'd be good here. The properties of the cable means it's flexible and transmits light down the cable with little loss, and it's also pretty cheap. Light exiting the cable if reflected, because of the material properties, so it bounces down the path easily. A duct with mirrors doesn't have this property. You'd have to make sure the mirrors are angled correctly, and it'd only work if the light is collumated, so it's all traveling parallel and not spreading out.
Fiber obtic cables, again, are just transferring light. They don't have an on or off state. It's just light. It isn't just used for data transfer. It's just good for that because it's flexible and transmits light with little loss, and little spread. Light that comes in one end almost all comes out the other end, no matter how it bends on its path. This also means it's great for something like transferring light in a building. You don't need perfectly aligned mirrors that would decay in effectiveness with dust. You just run some cables from a skylight to where you want light, and it'll come out the other end.
You can transfer data without fiber optics. It can be done with mirrors, just like your idea. The problem is that this system runs into a lot of issues. It is only done for data transfer in very specific circumstances, where you can't have a direct connection usually, because of this. The same issues would happen here (with a little less concern, because we don't care about data integrity, only quantity). All this is to say, fiber optics are great for transferring light where you can't have a straight line, not just for data.