Underpowering headphones isn't about volume level. They can get very loud, but they won't sound their best. There's just not enough power to let the drivers function as designed.
DACs and amps do change the sound of headphones, and it's not an issue of neutrality, but presentation as a whole. Underpowering headphones can make their desirable traits turn into the opposite.
As an example, the HD 6XX sound stuffy and dead on the DACport HD on low gain, simply due to being high impedance and requiring more power. Amps deliver different levels of power according to impedance.
The HD 6XX are plenty sensitive to make decent sound with little power, but that won't yield their best performance. The drivers have an impedance of 300 ohm, but that number is frequency-dependent and can increase much higher as music plays.
However, once the DACport HD is set to high gain, the HD 6XX are totally transformed to the point that they sound like different headphones, because electrically, they're functioning according to their design.
The bass gets tighter, the mid-bass no longer bleeds into the mids, *and the highs get sharper and smoother with much less "grain". The separation and layering increases, and also the width and depth of the stereo image.
It's the same DAC/amp, but the power delivery increased, so the drivers are behaving in a different, optimal way. You said you know you're underpowering the HD 6XX, that it gets loud enough, and that you don't think the sound signature will be altered by the DAC and amp.
If you don't have hearing damage, you will instantly hear the enormous change with the DACport HD on high gain. You will even be able to enjoy them at a *higher volume comfortably, because clean power yields less distortion. It's not an opinion of mine, but just plain science. 😎👍