The modern flash isn't firing earlier to enable higher shutter speeds--it's going into an entirely different mode, a fluttering or longer sustained burn than the 1/1000 or so normal electronic flash blast. It's not about timing the blast; it's about making it longer so that the narrower slit of the higher shutter speed receives illumination for the entire time it is passing in front of the sensor.
The normal sync speed for any camera is the highest speed at which there is still a point when the whole sensor/film is exposed, when the light fires. One curtain of the shutter opens, the flash fires, then the second curtain follows to close the shutter. Higher than that, the shutter becomes a smaller slit traveling across the sensor/film at the same rate as at slower speeds, and the light is limited by the width of the slit. If you fire a normal flash at one of those higher speeds, all you get is the spot on the sensor that the open smaller slit is over at that instant--usually at the leading edge of the shutter travel, smaller and smaller the higher the speed. I don't think modern digital cameras will permit you to make this mistake, but old film cameras will, however, so when your camera is empty, you can open the back and fire off a few flash exposures at higher speeds and see for yourself through the back what's going on, watching for the flash-illuminated slits. While you're at it, you can observe how FP won't do the job.