Well, I've said most of this in other places on the forum, but I have to get 1000 somehow, right?
I think the HD600 IS overrated in comparrison to other headphones, given their cost.
Among the flaws of the HD600 is a nowhere near flat response (it's all over the place), relatively low resolution compared to other headphones like the W2002, recessed vocals and (here it comes, wait for it...) sloppy, muddy bass.
I find these characteristics to be true from most amplifiers. However, I do agree with the general sentiment that the HD600 gets better as you get better amps. I was considering selling my HD600, which I had already heard with the Headroom Max, until I heard it with the RKV. Finally the bass tightened up and more extension was present. In other respects, the benefits were similar to that of the Max--cleaner, slightly better resolution, the recessed vocals feeling less underpowered (but still imaging too far back on stage for my taste).
For me, the absolute entry point to HD600 is the MG Head. If you're not willing to fork out at least enough for that, definitely look into a different headphone. Even with the MG Head, you'll find the bass sloppy but the sweet midrange almost makes up for it. If it's bass you're looking for, you want at least an RKV if you're going to get the HD600.
If you're going to get something like the Max and tight bass is the priority, I recommend the Beyerdynamic DT931. In many other ways, I think the HD600 is preferable but the 931 does seem more accurate and the bass is definitely tighter on that amp. It doesn't quite have the extension but it's pretty close and it's MUCH more refined on that amp.
As for music stylings, I can only guess that you'd take less issue with the recessed vocals on genres that have less vocals. I'd also guess genres that have less refined bass would be easier to appreciate. I'd suppose these reasons are why people think the HD600 great for classical. With the HD600, the trebble seems to dip in places that are often harsh with poor sources making them forgiving especially with classical. Bass is present in classical but rarely demanded to be precise or tight like we demand for electronica, punk and rock bass guitars.