Well, for future reference, yes, using a political term like "white privilege" that was invented to malign an entire race of people is, in fact, offensive and has no place here.
The problem here is that you've managed to let yourself get so spun up about race (by people for whom it is part of their business model, no doubt) that you can't even ask a question like "Is there a way to mention or include race when the profile is created for people of color?" without burying it deep within the responses to questions.
This is a question with a straitforward and obvious answer: "Yes, there is a way - just mention it." It's the same as for anybody who is from a special or unique population embedded within a different population. You could/should bring up if somebody is Jewish, or Amish, or German-speaking - whatever (if in the US). There are certain customs, traditions, and other implications (involving their interaction with the surrounding population) that make it important to tell other researchers what's going on. It effects the research, and it's something the descendants will be interested in knowing about.
This should be obvious: it's not the least bit racist, or "sensitive", to identify a black person as a black person, if it's relevant to the subject at hand. For something to be racist you need to make broad generalizations (generally negative ones) that because someone is of (whichever) race that they therefore are (a certain way). If such a thing is going on here, it is within your own mind.
If such a profile were in Nigeria, you obviously wouldn't bring that up - instead you'd bring it up if the person was of European heritage, etc.
As to the problem of the census recording such information, they have several terms they used, and you just need to become familiar with what they commonly meant. If they're unreliable as far as getting it right, well that's because it's the census.