Your task is not the same as classifying MNIST images. Images of the same class look similar, while there are numerous quite different board positions, which can be labeled as win or loss.
If your input data is 2-dimensional, then probably it would be better to make each sample in the shape of n_squares x n_figure_types. Imagine that you are one-hot encoding chess pieces for each of the 64 squares of the board.
Besides, you should somehow distinguish black and white figures. Either the input data should be 3D (square, figure type, color) or the number of categories in OHE should be twice the number of types of chess pieces.
Obviously, you should try to use some layers that can make sence of the relative positions of the pieces on the board instead of flattening the input data.