It is a great question you pose, and I wish I had a great answer. I actually really like how my 250ohm dt880's sound with my crack, albeit slightly less than I like the pairing with my hd650's. What I find so puzzling is that I also really like the dt880's with my O2, but I don't like my hd650's with my O2 much at all. The O2 clearly and understandably excels with high sensitivity orthos like my AD's (and at unity gain, with my Westone 4's), but the dt880's and hd650's are both high impedance phones of similar sensitivities that one would tend to assume would behave similarly. Aside from the extra 50 ohms, which is going to mean the hd650's are getting a little less power than the dt880's, my best guess is that it comes down to sound stage and spatial cues.
The dt880's are a little more spacious but less coherent sounding to start with, and I think the O2 brings that under control and focuses the sound. The hd650's, by contrast, are really coherent and focused sounding straight out of the box, and a little less spacious in terms of sound stage. The crack, and other good tube amps I've heard, make the hd650's sound a little more expansive and involving in a very good way (at least to my ears) - presumably something to do with the impact of even order tube distortion. When I plug them into the O2, their already very coherent focused sound is left as is, and I don't perceive them as sounding much better, if at all, than out of the jack on my Marantz receiver. As 300 ohm headphones, they also don't really benefit from the superb noise floor on the O2 to the same extent as lower impedance headphones.
I still haven't heard the GS-1, and I would love to be able to AB it with something like my mainline or a zana. My understanding is that KG's designs don't really fall within the "wire with gain" philosophy and actually add a hint of warmth to the sound of my headphones. Unlike the O2, my impression was that the GS1 and other dynalo derivatives don't rely on the same degree of global feedback as "measurement first" amps - which I have also read can cause interesting disproportionate impacts on very high order distortion. Although extremely low in magnitude, I wonder if the global feedback issue isn't tied into the reports of subtle "glare" that some listeners seem to perceive with the O2. Obviously, this is well above the threshold of fundamental frequencies, but high order harmonics are the controlling factor in our perception of timbre...
Also - thanks for your comments regarding benchmark and the significant downsides of going balanced. I had read Elias Gwinn's debate with AMB on the balanced issue a while back and found Gwinn's position very persuasive. (http://www.amb.org/forum/benchmark-engineer-on-balanced-v-unbalanced-headphone-amps-t326.html). (I think you also raised a very interesting point regarding benchmark's lack of specificity as to the amp modules included in their products. In a sense, you have to wonder if they are looking to avoid the sort of scrutiny that they are directing towards competitor's products.)
That said, with the exception of Pass Lab's recently expired SuSy patent, it seems like there is a lot of truth backing up the argument that balanced amps have more cons than pros. To recap: unless you need the extra power to drive an inefficient transducer, push-pull or, alternatively, bridged amps generally add noise and distortion (excluding, in push-pulls, common mode distortion - which is really only relevant to long cable runs). They also raise output impedance/reducing damping (theoretically bad but which may or may not be to the tastes of the user at issue). Most notably, they cost about twice as much to build. On the flip side, you get double the voltage and double the slew rate, but the latter is believed by most to be inaudible in the context of any competent amplifier.
I don't have the technical background necessary to fully appreciate the why of it, but if you look at Pass Lab's technical summary article, you will note that Pass essentially agrees with the Benchmark guys in so far as suggesting that the 3 traditional balanced approaches either perform worse than, or at best, equal to, ordinary 2 channel single ended designs. https://passlabs.com/articles/super-symmetric-amplification SuSy amps, as discussed in the link, are evidently a different matter.
I have read that some people think the KG designs slightly warm, but I never heard the GS-1 that way. The GS-1 has much more information in the upper registers and has a significantly higher resolution than either my M^3 with 637/627s or my Woo3 with a Cetron power tube . I would classify both the M^3 and Woo3 as having warmish presentations relative to the GS-1 but not the holographic effect the GS-1 has. Curiously, the M^3 (637/627) and the Woo3 (Cetron) have extremely similar sound signatures and I have found my 650s sound almost the same from either one with maybe a hint more bass from the Woo's second order harmonic.... maybe.
Regarding the O2, the GS-1 can play louder than the O2 but I don't think I'd want to bet on my ability to distinguish between the O2 and the GS-1 at moderate listening levels with my T-1s (dac used was a North Star 192MKII with either a Blue Circle Thingee or a HiFace transport, ... the HiFace having more resolution and a hotter treble than the BCT and the Thingee having less resolution but a better tone ). Unfortunately, once I discovered no apparent JND between the O2 and GS-1 with the T-1s, I never bothered to compare extensively or further with the 650s, '04/880s, or the 701s I almost never use.
These days I use the O2 with a laptop, a Pico dac and 880s when I'm away. When I am at home, I use the same laptop but with the HiFace/North Star/GS-1 and either the 650s or T-1s depending on mood and program material.
What I did notice recently was that when both the 600ohm T-1s and the 300ohm 650s were plugged into the GS-1 at the same time, the T-1 played louder than the 650s.
Now I'm thinking about another direction.... Harbeth P3ESR or KEF LS50 and some kind of compact dac-amp like a Peachtree Nova.... maybe