Thanks for the question Daniel.
I might be misreading your question, but I think you're missing the "each person gets the same color as the person seated directly across from them". Because of this, any way we cut the circle in half, the set of people on one side are always directly across from the set of people on the other side, so both halves must have $4$ white and $4$ black cards. In your example of all black cards for one half, then all the cards on the other half would also need to be black, which contradicts the $8$ of each color portion of the problem.
Let us know if you have any other questions. Thanks!