I have heard the stock HD800 out of the HDVD800, and also the LCD-2 and LCD-3 out of the same rig.
Normally, I'd prefer the LCD-2, and to a certain extent, the LCD-3, but out of the HDVD800 specifically, I'd pick the HD800 over the other 2 any day. The LCD-3 was somewhat comparable, but the LCD-2 couldn't keep up at all.
On the other hand, if it's out of a Schiit Lyr with a Schiit Bifrost, which I have also had the pleasure of listening to, the HD800 would be just okay while the LCD-2 would rock.
So I still think synergy, or at least the very subtle differences between amplifiers can make or break the sound of a headphone.
Being able to make something sound loud and being able to make something sound good are very different in my opinions.
I have owned the E12 at one point.
I currently own the HE-560.
I have heard the LCD-XC and HE-560 out of the E12 very recently. The pairing is suboptimal to be polite... and in fact, I wouldn't even say it can drive the HE-560 to "satisfactory" level. The LCD-XC is loud out of the E12, but just that. I have heard it out of better amps, and it does sound better than with the E12. The HE-560 wouldn't even get loud, and at high gain, it actually started to distort a little at lower frequencies.
And yeah, some headphones are just that picky with source components even when they don't seem so. But that doesn't mean the headphone is "good" per se. It's just that it's picky. I am starting to think the Z7 may be conforming to this pattern.
Beyond that, if you want to go into the semantics of why different electronic contraptions (amp, DAC, or DAP, etc...) should sound different, I believe we can continue the discussion in the sound science section.