Although it is more difficult in 3D, it is still a very solvable problem, that won't work always, but in 99% of cases if the camera angle is not very low.

Indeed, the checkerboard itself gives valuable information about the pose of the checkerboard, it is even used for calibration in multi-view geometry: https://en.wikipedia.org/wiki/Chessboard_detection

I wonder how much easier it would be if you narrow down the likely set of pieces for each square. There are obviously some positions for some pieces that are invalid (bishop on the wrong colored square), but there are probably a lot of other positions that are so uncommon that they could be discounted.

Standard object detection can't work here?