Correct me if I am wrong....... (famous last words)
This is not necessarily something intrinsic to DSLRs but is a fundamental issue with a focal plane shutter, and applies to Digital or Film.
I will try and explain although a Google search (substitute your favorite search) will I am sure turn up a more lucid explanation.
Most handheld DSLRs (maybe with the exception of the Leica S) still use a focal plane shutter which allows light to hit the sensor by opening a (first) curtain, and then at the end of the exposure shutting a (second curtain). Due to mechanichal factors (materials strength, inertia etc.) there is a maximum period of time when the WHOLE area of the imager is exposed to light at the same time. Above that period of time (or shutter speed) the second curtain follows the first curtain so that (in effect) the light (of the exposure) scans across the imager.
Focal plane shutters that ran horizontally across the image plane (Leica M style) which used a treated cloth material for each shutter curtain were limited to a maximum synch speed of around 1/50 or 1/60 second. I recall talking to Leica folks in the 80s when I was in the business and their view at the time was that that speed was the best trade off between consistent accuracy, durability, and quietness. At the time Vertical run focal plane shutters (and remember because of the rectangular nature of the 35mm frame, the vertical shutter blades had 2/3 of the distance to travel) eventually crept up to 1/125 second and stayed there until the 90s when IIRC they hit about 1/200sec
The shortest amount of time when the WHOLE imager area can be exposed at once is therefore the highest flash synch speed. in the mid 80s some companies (Olympus comes to mind) came up with an innovative way to get around this by pulse firing a strobe (at obviously reduced power) so that strobe light would be evenly distributed across the film at higher shutter speeds than was normal at the time.
This limitation is one of the reasons why medium format camera systems (where the size of the shutter would be greater, and thus maximum synch speed lower) tended to use a leaf shutter. The leaf shutter rather like the lens aperture has concentric blades and can close and open fully, thus at any speed you can always get flash synch (although I cannot recall ever seeing anything higher than 1/800 sec).
The advantage of the aster shutter speed beyond freezing action (or the inverse, preventing camera shake) was to control the amount of ambient light in the composition. e.g. if you want to darken the background when using a strobe.
The whole point of this long winded explanation is that I suspect that since maximum sync speeds in focal plane shutters have not got much above 1/250 sec. in the last 20 years there is probably a major physical limitation to doing so. I am willing to bet if camera company X came out with a camera that allowed the use of all strobes at 1/1000 sec just about every wedding photographer would buy it!
The question I have is if it is theoretically possible to do this by using a slower shutter speed like 1/250 sec and "faking it" by taking a 1/1000 sec sample of the image created during the 1/250 second exposure. I see that you would lose two stop of illumination overall but it appears that between 100 and 800 ISO most modern imagers have sufficient latitude?
Lastly, why, if you can get 1/250 on full frame can you not get higher on a crop sensor, surely the shutter run distance is shorter, and so the synch speed should be higher?