Many of the CD players that employ built-in "variable" volume control actually implement it in the analog domain. As stated in the Accuphase verviage, some do it in the digital domain.
I've owned many different CD players with built-in volume control. For the most part, the benefit outweighed any potential down side. In my case, I have a single source system. Why spend $1000-$4000+ on a pre-amp when the CD player has a decent volume control? The cost saving alone allowed me to afford a much more expensive (and better sounding) player than I would have purchased if I also had to drop a bundle on a pre-amp.
BTW, there is nothing keeping a manufacturer from putting a very high-quality analog or digital output stage directly into the CD player. If Accuphase has a great digital pre-amp using 48-bit DSP, why not use that one in the CD player instead of the "admittedly" piece of crap that they use? If they already have the technology, why not use it? I'm gusssing it is so they can sell $5000+ plus pre-amps in addition to the $5000+ CD player.....
For me, CD direct is the way to go. It saves me a bundle of cash and it frees up shelf space. It also provides the least possibilty of the signal getting mucked up between the source and speakers. Some will say that dynamics suffer with no pre-amp. Well, I haven't noticed it in my system, my room, with my musical preference. But I don't listen to large scale classical, so that might be a different story.....