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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.
The reason fiberoptic is better than a mirrored duct for data transfer is because you can pack say 32 or 64 fibers into a cable, that means 32 or 64 points of light that are either "on" or "off", creating a 32-bit or 64-bit word size and enabling data transfer. You can't do that with ducts.
There is an on and an off state. Either the light is on or the light is off. That's how it translates into binary. Literally that's what binary data is: a single data-point is either on or off. Put a bunch of them together to create words with 2^n^ possible combinations per word where n is the number of datapoints. For electrical data, that's the voltage level of one wire or bus lead. For fiberoptics, it's each individual fiber. It's either on or off, that's the whole point.
The mirrors don't have to be angled precisely. If you take a cylindrical tube, and make the inside a mirrored surface, then all light traveling down it will continue traveling downward as it bounces. The only time the angle matters is around turns, but that's easy enough to angle correctly.
It also doesn't need to be columnated, but the thing about the fish-eye dome is that with a flat lense on bottom, it does output columnated light from the wide-angled light it receives. That's how convex defraction works.
And dust wouldn't be an issue if your tubes are sealed.
Fiberoptics would technically work, but it's more material than you need because it would require running fiberoptic cables everywhere instead of just using hollow chromed tubes. Also, the quantity of light it can receive and transmit is limited to the thickness of the cable.
Fiberoptics are great for high-speed data transfer because of data-integrity and the fine-pointed nature of the fibers. But they aren't ideal for moving large amounts of light where precision isn't needed, e.g. enough natural daylight to brighten several rooms.