Well, not quite true. The headsets on the older Schwinn's had a cone as the thread-on portion and cups on the head tube, rather than the more modern setup of a thread-on cup and top cone race. As one screws down the cone it is quite easy for it to start moving the balls out of place by catching the grease as it turns, before the headset is firmly together. That is exactly the reason I developed the method I laid out, and if the OP follows my instructions he should be able to assemble it properly.