Without a schematic the horse games are tricky, you will need one I believe.
There are a few 3 way switches (one on the bank tilt, game and reset relay I believe from memory) that need to be correct on the power return line to the transformer or this happens. On an older 60's game the flippers are initially activated only when the ball goes through the gates into play. This is to assist the ball counter stepper operate correctly.
Good luck.