I think you're wildly missing the point.

When someone asks to see a "white family", they are not asking for a family with skin of a certain shade. They're asking for an image in which our pattern recognition identifies in their clothes, posture, hair style, and facial features that they look like people who could appear in a soap ad in the 1950's. That they look like people who feel totally welcome in their society. They live a certain lifestyle. Simply changing color is the point of the problem. Koreans look pretty white in skin color, but they have other facial features that communicate that their parents or ancestors father back left the land of their birth and traveled to the US likely after 1900. Additionally, based on their dress some people might look at an image of a family with a Korean dad and say, 'Great, that's a white family', while others would say, 'Why did the model generate this? I asked for a white family.'

There's a world of context that our current racial terminology can't capture because it's not suited to our modern understanding of culture.