I’m wondering what all the above has to do with the GS/PS2000E.
I remember being okay with the sound of a Dragonfly Black some years ago, but this was well before I got “serious” about audio so I’m dismissive of my own impressions back then. In my (admittedly limited) personal experience with good gear since then— yeah, unless you have a proper line out function you’re introducing the kind of distortion even self-professed tube fans such as yours truly find unpleasant. Think hard clipping and fuzz; it’ll always be bad, though whether it’s tolerable or not depends on other components in the system, listening skill, persnicketiness, and quality of music being listened to.
Speaking as someone who listens to an unhealthy amount of mid-2000s pop rock/punk/pop besides all the more “sensible” selections, the degradation is pretty noticeable even then when you’re double-amping that way. All the Dragonflys (pluralisation of a product name that’s actually a common noun always gives me headaches) seem to be good value products, just not designed for pairing with another amplifier.