I forgot you have an HD600. That was my first audiophile class headphone and I'd still own it if it didn't break because it's a keeper. You're right, this headphone scales very well and should have no problems revealing the qualities of the DAVE so all the more interesting that you preferred your Nagra CDC's built-in headphone socket to the DAVE when it really should be no contest. Having now heard my HE-1000 through the headphone port of the more capable Nagra HD, my opinion is that I easily prefer the DAVE. My only thought is give a well broken-in DAVE another shot but undoubtedly, you have to go with what your ears are telling you.
Regarding your interconnects, obviously this is not what is holding back your system.
Regarding how a DAC should sound and some may disagree but my feeling on this is that it should only sound one way for each of us and that personal preference should have less to do with it. Regardless of whether you are a novice listener or a seasoned and discerning audiophile such as yourself, a DAC is either faithful to the full virtues of the recording or it is not. The music either sounds real or it does not and while your headphones and amp have some role in this, I am finding that the DAC perhaps has the biggest say in it provided that you are starting out with a good digital file. With DACs, I also don't buy into the idea that you have to have some proverbial golden ear to appreciate its finer qualities because as humans, even if age has hampered our hearing to some extent, we each have an innate qualitative sense of correctness and what sounds natural even if we can't articulate it as well as others. As the old saying goes, it's like pornography, you can't quite define it in words but you know it when you see it, or with music, you know it when you hear it. If it takes hours or days, then its probably not there.
Of course, there is the topic of coloration and preferred style of presentation and this is not the same thing as the whether music sounds real or not. There are some that prefer the fast, focused and full bodied acoustics of the Alice Tully Hall in New York compared to the more bloomy presentation of Carnegie Hall. There are also some that prefer acoustical music in an intimate setting compared to amplified music in an arena and sometimes, this has more to do with your mood on a certain day, but this is why we own multiple recordings of Mahler's 8th symphony performed in different venues or why we roll out the 300Bs from our tube amp and replace them with 2A3s. It's why some of us own a more laid back Audeze LCD-3 in addition to a more analytical HD800 or a basshead TH-900. In my view, coloration belongs to almost any other piece of equipment in our audio chain but not the DAC. The DAC just has to be faithful to the recording. It has to sound analog.
Regarding the specific virtues of the DAVE and what makes it so special, there are already so many people who have chimed in on this but I will sum up in one word what makes the DAVE special for me -- "timing." Time resolution impacts almost everything. It impacts tone and timbre. It can tell you that the lead violinist is playing a Guarneri and not a Stadivarius. It tells you where your woodwind section is in the orchestra relative to the horns and and how many rows of woodwinds are present. Because higher pitched voices reach your ears sooner than lower pitched ones, you are aware you are listening to a thousand voices in the choir. When time resolution is there, then music is an onion, full of layers. As I've learned to appreciate the qualities that DACs provide, the one quality I have found that has been most difficult for a DAC to recreate is space. In the same way that a saxophone can only sound good when it breathes, music has to be able to breathe to sound good. We talk about soundstage width and height but without depth, then the music is flat, and its not something you can convincingly fix by adjusting the reverb on an EQ . It is this singular feature that differentiated the more basic $10K TotalDac d1-dual from the more sublime $25k TotalDac d1-monobloc and it was enough of a difference for me to spend the extra money. It is this same special quality that has lead me to the DAVE and thus far, I have not heard another DAC do this better.