Yes it does

Makes a big difference spraying a thin coat of satin black on it; cuts the temp in half as felt by hand. And this is before being in contact with water.
Delta T does not play a part; it's radiation, not conduction. And it's not insulated on the top; just the sides and bottom.
Plus, it's working on multiple units
Your position contradicts known physics and the established performance (science) of heatsinks. Respectfully, you can't just toss out a scientific term, a misapplied definition and a personal observation and succefully contradict the laws of the universe... I am not looking for a personal fight here, but rather simply trying to ensure that we don't mangle the laws of the universe in order to promote a product or idea.
Is your heatsink capable of moving enough heat to serve the purpose? Maybe, but it is not because you painted it flat black.
black-body radiation - Anything in the universe that has a temperature over absolute zero RADIATES energy (in the form of photons). That is, everything in our known world is a black-body radiator in the context that you used the term. I am not painted black, but can be thought of as a black body radiator, as can your raw aluminum heat sink or your left eyeball or your cold beer can or an iceberg.
Stefan's Law (Stefan/Boltzmann equation) - Defines the quantity of black-body energy radiated from an object (our heatsink in this case). The law relates the emissive power to the area... It says that a body radiates energy at a rate proportional to the 4th power of its absolute temperature times its area....
Notice the 10<SUP>-8 </SUP>, that constant is very tiny... so either your heatsink must be VERY hot AND the surroundings very COLD (delta T) or the heatsink must be the size of a battleship if it is COLD (low emissive power).
Black Paint - The COLOR of the paint being BLACK has nothing to do with black-body (see the tiny constant we are multiplying by). From an emissive standpoint, the paint color is not really relevant. What is relevant about it with regard to emissivity? Its chemical composition, but even that matters little for our given use scenario (temperature and size). I ask again, is your heat sink aglow?
Again, MOST of the energy loss of the heatsink is due to contact (CONDUCTION) between the molecules in the air and the molecules at the surface of the heatsink. Lets say 85% to the 15% emissive (radiation) transfer at the temperatures we are working with. If we add a layer of paint that increases emissivity by 20% but reduces conduction by 10% we are going in the wrong direction....
All that said, raw aluminum is not ideal at either conduction or radiation, so anodizing (or painting) the surface may help in both departments (color does not matter) and will also serve to protect the aluminum. If it is paint, the conductive properties of the paint are more important then the emissive properties (see law governing black-body transfer above). A finned or rippled surface will be better than a smooth surface, etc. Our major (only) concern is conductivity, be it driven by passive or active (fan or convective) air movement.