Class D amps get away with "lighter" power supplies because they are filtering at many KHz instead of the 60 -120 Hz of linear supplies. Thus, they can use much smaller capacitors for a given current or power rating.
Look at PC power supplies, for example, which use a similar switching technology. They currently build 1000W-1500W power supplies for computers that fit in a box 6"x6"x8". And these run at 100% continuous power, as opposed to audio systems where the average power is about 1/10 of the max at full output.
That said, I have always been suspicious of the audio performance for such amplifiers, except in subwoofer applications. Maybe I'm old fashioned, but if Class A/B is inferior to Class A, the class D has got to be inferior to Class A/B, simply because it operates completely out of the linear range of the output devices (hard on to hard off).
I'm not sure why the PC type of switching supply is not used more for audio (actually it is in car audio, but only there), but I suspect the audible side effects of such supplies are intolerable when compared to old fashioned linear supplies with thousands of microfarads of capcitance.
They are the cheapest way to make really big power, but I am not sure if they are the best.