AFAIK the only Amiga that would have come with a DSP in a default configuration is the never released A3000+ which used an AT&T DSP3210. Prototypes of the A3000+ existed in 1991.
The major contribution of the original Atari STs to music were the built-in MIDI ports, which made it easy and simple to connect digital synthesizers, keyboards, etc. The music capabilities of a regular ST were rather constrained due to its outdated AY-3-8910/YM2149 sound chip (the SID in a C64 was more capable). Only the STE introduced PCM audio.
I guess someone more knowledgable about Amigas than me can add some facts about its sound capabilities.
[1] https://stcarchiv.de/stc1993/10/interview-atari-bob-gleadow
The Amiga, on the other hand, didn't have that (or even a synth chip), but it did have the Paula chip, which gave you the ability to play four channels of samples, which got you (generally) 8-14 bit/22 kHz output on the original machines. On that side of the fence you got a lot of trackers - four channel sequencers - to take advantage of the digital audio.
Not influencial in the way the Cube is described here, as the platforms were a class of apps was developed in, more used practical applications.
It was the direct ancestor of today's VST/AU/AAX/etc synth and FX plugin market.
I really disagree about it as a direct ancestor of the contemporary plugin API. These have always run on the host CPU, not dedicated DSP chips (excluding of course Digidesign's original ProTools model, but that was DSP farm rather than just a chip).
And the sort of DSP that was being done on the Cube DSP was being done before the Cube too.
That's like saying "computing in the 80s was all about the home computer market, if we exclude the PC". ProTools was the biggest name (and quite close to a monopoly for pros at the time).
ProTools had a separate box that had the DSP farm in it. There was nothing particular unique about that (at the time) (though running a DAW on it was definitely novel when they did it).
When NeXT was folded back into Apple, the audio classes eventually became (more or less) the AUs in MacOS.
An abstraction layer made it possible to run natively, or on internal DSP, or on external DSP. Aside from Digi, Waves, Universal Audio, TC Electronic, and Focusrite all made external processors that used the AU/VST interface. (UA still do, although recently they - at last - also started offering native versions of some of their plugins.)
Waves in particular, who I worked with in the 2007-2014 era, did not do this until Linux got into shape to allow them to build plugin/DSP servers.
And it was used in serious modern music, as practiced in the various modern music departments on the best universities, even when they did have the big money. In Graz they had this MAX/Next setup from IRCAM and simulated opera singers, in Paris at the IRCAM which did a famous movie with it, the Neitherlands, Gent and Montreal.
They continue to work on PureData, but it began taking over the industry with Ableton then. And with all it free clones.
You're probably right that approximately noone was making rave and jungle on NeXT machines, but on the other hand it's true that approximately noone was doing academic electronic composition on Amiga or Atari, which is really the tradition that this is referring to. And these were mostly separate worlds for another decade.
Aphex Twin used Amigas, not NeXTCubes.