@The Phenominal it could be because in a way they did at least in the originaltimeline. In the original timeline line the "bad guys" aka black prices made the 1st move killing his sister they schemed and ploted and use various dirty tricks to destroy his family.
So black and white not only represents him as good and them as evil but also shows how black has/will break the rules of the game to get an advantage aka going 1st when they should go 2nd. So I don't think its to show he is now controlling the game but showing how he will use his future knowledge to counter they cheating and respond in the best way possible to give himself the best chance at victory. He has essentially accepted that he now has to take control of the flow of the game back from black who stole it by taking the 1st move... or ya know we are both over thinking this and it's just a simple mistake like you mentioned as a possibility.🤷♂️
So black and white not only represents him as good and them as evil but also shows how black has/will break the rules of the game to get an advantage aka going 1st when they should go 2nd. So I don't think its to show he is now controlling the game but showing how he will use his future knowledge to counter they cheating and respond in the best way possible to give himself the best chance at victory. He has essentially accepted that he now has to take control of the flow of the game back from black who stole it by taking the 1st move... or ya know we are both over thinking this and it's just a simple mistake like you mentioned as a possibility.🤷♂️