Painting or anodizing definitely improves radiation dissipation, but it doesn't have to be black. The colour in the visible spectrum has absolutely no correlation with the colour in the IR heat spectrum of a heatsink. So you can get any colour you want, as long as it is "black" in the IR frequencies. Most standard paint of any colour is fine to use so fluorescent green is as good as black. You don't even have to paint or anodize it. Stick masking tape or electricians tape to the raw aluminium heatsink and you also get radiation cooling close to a black anodised one.
This is not the same as choosing the paint colour for a car as there is a lot of energy from the Sun at much higher frequencies then heatsink IR emission, and definitely a black car gets hotter then a white one. I think the Sun must be a bit hotter then a 100 deg C heatsink.
Unpainted aluminum - shine or rough - has lousy radiation and also doesn't absorb radiation from other heats sources.
This fact is really useful. If you want to have fan cooled heatsinks in a case, you want them unpainted, so they aren't radiating heat inside the case. Also if you paint the outside of a heatsink but not the inside, it will only radiate significant heat to the outside. Nice!
On the other hand external convection cooled only work with a height up to about 100mm. A 200mm high heatsink only gets about 50% more convection cooling as a 100mm high one and over 200mm, you are just wasting expensive aluminium for no good reason. So if you need a very tall heatsink, forget about the fins. Make it a flat sheet, paint the outside surface any colour you like, and get rid of almost all the heat by radiation.