I always thought that the heavy aircraft want to accelerate above 250kts so that they can clean up and fly with flaps/slats retracted.
They sure do in many cases.
The maximum speed for a turning departure in a CAT D aircraft is 290Kt. Therefore once the 250Kt limit is removed subject to airspace limits and company SOP's, accelerating to 290Kt in the absence of a lower limit will not cause a problem.
Here are the standard max speed for a turning departure;
CAT A 120Kt
CAT B 165Kt
CAT C 265Kt
CAT D 290Kt
A procedure can within limits require a speed lower than the above for a turn but that will be specified on the chart.
One can see that it is only CAT A and B that can not fly at 250Kt on a turning departure procedure appropriate to their category.
Taking the CAT B case since many of the smaller citation jets etc fall into this category. When departing somewhere like Heathrow, etc they can happily accelerate to 250Kt because the SID is constructed for higher categories. However, they can get into trouble when this becomes the norm / a habit and they try to do the same on a departure which is restricted to CAT A and B aircraft only.
Like I said, these are speed limits that ATC can not absolve the pilot from complying with.