Yes, the flash duration determines the speed of the exposure and not the shutter speed. With studio strobes and a focal plane shutter, if you set your shutter at a higher speed than the sync speed of the camera, only part of the frame will be exposed, and part will be black usually. If that's not the case in this situation, then it is likely that the flash is weak compared to the ambient lighting.
The way that high speed on-camera flash works is to flash repeatedly as the shutter slit moves across the frame, but it does so usually at the expense of power, so you can't usually get the maximum power out of your flash this way.
If you are using a handheld flash meter, it should measure the flash output independent of the shutter speed set on the meter. Try setting the meter at various shutter speeds and firing the flash, and you should get the same f-stop no matter what the shutter speed, unless it is metering flash+ambient, and then you might get a smaller f-stop when you are using a fairly long shutter speed.
A powerful studio strobe has a fairly long flash duration at full power--maybe 1/500 sec. That's why for dance photography, one might use several heads at low power, rather than one head at high power. Your heads have a flash duration of 1/1600 sec. at full power, according to the Buff website. At half power they have twice that speed, etc.