I think you will find that there is more than one reason for a drag required message.
According to the Honeywell FMC guide for the 747, "The DRAG REQUIRED and THRUST REQUIRED messages are displayed to advise the pilot of speed changes required to maintain the descent path.". They don't give a lot more info about it.
I had one yesterday on a different Boeing but it was not because of airframe overspeed concerns. It was only to meet a speed restriction on a transition route at about 4000 feet. So if nothing had been done, I don't think it would have gone to VNAV SPD, it just would not have met the speed restriction without speedbrakes.