The car does not keep moving backwards because the backwards force only exists while there is relative motion between the brakes and the wheels (the friction force). As soon as this motion stops, the force abruptly stops existing.
So, if the force persisted, you are right to think it would deaccelerate the car to a halt, and then keep accelerating it backwards. But this force is not an independent variable: it depends on the motion itself. The car stops, the force stops, and the resultant is no longer negative.