I think, in the US, skin pigmentation has historically acted as a metonym for the social divide between European-Americans and African-Americans. But, it would be a mistake to take the metonym too literally – and people who are neither have always been somewhat out-of-place, and as the US becomes more diverse the number of people who don't belong to either side continues to grow. Some Indians have darker skin than some African-Americans do – would you expect therefore, that when those Indians immigrate to the US, they'll encounter worse treatment than lighter-skinned African-Americans? I don't think anyone believes that is what actually happens. That's not to say that they don't sometimes experience discrimination to a degree which lighter-skinned Indians do not (whether in India or in the US), but it is an overly simplistic model of how US society actually works to suggest that there is such a simple correlation between one's skin tone and one's societal privilege.
Also, "white" people aren't all one skin tone. My wife and I are both of predominantly northern/western European descent, yet my skin tone is somewhat more "olive", hers is quite pale. In many Irish families with slightly darker skin, the darker skin tone is explained through the legend of Spanish ancestry (ship-wrecked sailors from the Spanish Armada) – I've heard that legend with respect to both my parents' families. Am I, as a slightly-more-olive person of European descent, discriminated against compared to paler-skinned people of European descent? Not as far as I am aware. But, that once again shows that there is no simple and historically universal correlation between skin tone and privilege.
> My argument may be misguided or even wrong in any sense of that term, but I do not think it is naive.
You are American? You seem to keep on making the assumption that ideas derived from American culture are universally applicable, even when more than one non-American tells you that they are not. How is that not naïve?