That could be true, that the hair is dyed due to mummification; however, I don't think it is very likely. The way dying (or rather, bleaching) your hair works is that first it turns a dark orange to lighter, to yellow (or blonde) and finally white.
Mummification took relatively the same amount of time, varying from corpse to corpse, and of course royal mummies had stronger salts and longer, greater care taken into the process of mummification. Now, there are royal mummies that have hair of dark brown, black, red blonde and white. Personally, I don't think it is due to dying of the hair, as it would mean that all mummies would have at least a lighter shade of hair than black or dark brown.
I think it depended on how acidic the natron was, not on how careful they were mummifying. That is why many mummies have that fake orange bleached look, but the ones that have darker hair had less acidic natron.
Egyptian people really aren't black or white. We have to consider how many different races of people came together. Upper Egypt has darker skin, because of a more Nubian background, and Lower Egypt has that Hyksos and Asiatic influence, along with kings beginning to add Eastern wives to their harems. When Amunhoptep III married Gilukhipa, daughter of Shuttarna II, king of Mitanni, she arrived in Egypt with 317 ladies in waiting.
And we have to take into account that the Ancient Egyptians never considered themselves black. They considered Nubians black, but they always portrayed themselves with a middle skintone.
The only "black" statue I can think of is the Osirid statue of Mentuhotep. Of course, the black seems to be an attempt at trying to portray him with the Osiris skin tone; blue or green. During the beginning of middle kingdom, the art was still rough, and it had been nearly 200 years since the arts had flourished in the old kingdom, so the statues artistic style is not up to par with the art that was produced at the hights of the ancient Egyptian Civilization.