Why is the Church still promoting the idea that black skin is inferior?
Because it's NOT referring to black "skin", it is a metaphor in relation to "purity", a.k.a. righteousness.
Similar types of metaphor is used all through the Bible/Scriptures (as the below videos show).
The ONLY issue with this particular metaphor is because it was a bit more "accurate" due to Lamanite inter-marriage with locals who tended to be "darker" in skin color compared to the Nephites who didn't inter-marriage as much.
However, this is not 100% correct either, because inter-marriage is the only thing that can explain Nephite population growth.
Thus, ultimately, this cannot in ANY manner be in relation to "race" or "skin color", but in relation to righteousness.
Some of I've stated is well explained in these videos.
Black & White Skins in Light of Ancient Garment Color Symbolism in Book of Mormon
https://www.youtube.com/watch?v=RQFev9IX-70
https://www.youtube.com/watch?v=O64PJgxEWhI
https://www.youtube.com/watch?v=_a0_nRgL9cE
https://www.youtube.com/watch?v=lzp70cBiLV4
https://www.youtube.com/watch?v=WGFhnuBsqxA
https://www.youtube.com/watch?v=tmi-Ya9ySY8
(Please ignore the eccentricness of the music and the speaker, but focus on the FACTS presented, they will allow you to understand scripture better, in relation to various "metaphors" (i.e. Color Symbolism and otherwise) being used to describe righteousness, or the absence of it.)
The problem here is NOT that the Book of Mormon is "racist", it's that Anti-Mormons do not understand the Word of God.
Darkness clouds their minds and souls in understanding Truth and Light.
The total lack here of awareness by Nelson of how his own church denied the most basic of priesthood authority and temple blessings to someone because of the color of his or her skin
Correction... Priesthood authority and Temple Blessings were ONLY denied to those considered of "African"
Lineage, including some WHITE individuals being denied it.
Individuals who were just as BLACK as any African black, certain groups of Central/South America, Natives of the Islands, those of India, etc.
were NOT denied the Priesthood.
Thus, it is 100% FALSE to say that those of "black skin" were denied the priesthood.
The policy denying the priesthood to the African lineage was entirely due to an interpretation of scripture, and due to a revelation of policy to Brigham Young. Given that the policy was lifted when institutional racism and with most people racism itself was finally gone from America in the mainstream, it's not a wonder why it existed in the first place, and why and when it was finally lifted.