The drag center and mounting point of the chutes is above the metacentric height of the bodyless car...when they deploy, they tend to lift the rear end of the car. This is overcome by the downforce generated by the body, but without the body...also, when the body blew back over, it lifted the car before the hinges failed.
Modern cars run smaller but deeper chutes to allow them to be mounted as low as possible. 70s cars like that had them pretty high up on the chassis.