I would guess the orientation of the magnet in some positions is not enough to magnetize the contacts of the reed switches.
The first thing I would do is test the reed switch vs magnet orientation. Start in 2D, meaning attach the reed switch to a piece of graph paper. Then by bringing the magnet in close to the reed, measure where the reed switched. Then rotate the magnet 90 deg and repeat. Then rotate the magnet in the Y plane repeat both the above.
Now rotate the reed switch 90 degrees. Repeat above.
You should be able to fine the optimum orientation.
Now try the reed on your board, put the chess piece at the edges of the position and see how close to the center it needs to be to work.