This is a smoke screen, they are still "basically" the same, with the same glaring problem, upper mid/highs phase shift, because of the "switching frequency noise filter" on the speaker outputs.This statement is false- they are not 'basically' the same at all.
Bruno Putzeys has shown that you can run enough feedback in a class D amp that it can correct for phase shift at high frequencies that would otherwise be the result of limited bandwidth. To do this the amp has to have more than 35dB of feedback. Any less and the phase shift will show up. In this way the bandwidth of the amp can be limited to 20KHz and yet no phase shift in the the audio band. Normally you need lots of bandwidth (usually 10x the frequency of concern) to prevent phase shift.
With traditional solid state and tube amps, this much feedback would likely result in oscillation due to phase margin issues. A class D amp can take advantage of this because the oscillation is welcomed- and is used as the switching frequency. This type of class D amp is known as 'self-oscillating'. The pulse of the switching frequency is then converted to a triangle wave, and compared to the input signal to create Pulse Width Modulation (PWM) for signal encoding. Its a neat trick- and then you don't have any of the issues described in the quote above.