I think the real answer is, "it depends." For one thing, being in a post-modern society that tends to hold there are no absolutes, then no one can say Spider-man has to be white. As long as you can tell a convincing story, then all is well. Fury as a Black man is a convincing story; Michael Clark Duncan (and the whole movie) not so much.
But I think there are way too many variables to make an absolute statement. Take the Wiz. The original is such a classic that even in this age of "let's remake every movie ever made". They haven't tried to remake this one. But with the Wiz they did reimage it. That served a purpose because it was a really different movie. The core remained, but the show stands unique. So it was creative, yet not.
Along that line, I think the switch is used, because mainstream is afraid to try and make really new stuff. Colorizing a few guys in a remake is safe money.
Part of this I think is Whites are constantly drilled that the only difference between Whites and Blacks is skin color. The goal being, race is no more important than hair color in a person's identity. (That's where the ST world takes us. Just wait till one writer realizes that the tech of Star Trek makes it possible for people to change skin color, hair, etc like dying one's hair today. Being Black in ST world isn't a matter of heritage, but choice.)
But all in all I think there is a danger that colorizing a character falls back on the premise that "white" is the default race. Saying you can turn a character from white to black without changing the character, makes it seem as if white is the universal core and Black is a one of several interchangable faces that can cover the core. The new Karate Kid isn't doing this. The new guy isn't Daniel with new skin. He is a completely different character. But take Pete Ross on Smallville. Nothing about him was Black. If they cast a white man to play him, not one detail in the story would have changed. The same is true of Kingpin in the Daredevil movie or Nick Fury (so far)in the movies. On one level it does provide diversity, but on another, it can suggest that being Black is "take white character, add Black" if you do it wrong.
My thought, if you were to do a Black Spider-man, then make him a new--but similar--man inside the costume. Just as in the new Karate Kid, he is a new guy who has a different name and is years younger. He is unique yet the familiar.
(Still, all of this means, play it safe with your product--use the familar. Original ideas need not apply.)