I comment this because one of the guys who loves the K701 with the V8 thought the GS-1 is an overly cold and analytic amp so maybe the OP would think the same.
I think the 701 are decent for the price and may work for many people with the right amp and source, but once you know "better" there's no way back.
You nailed it. A euphonic amp will add pleasant coloration to the sound, but won't remove the inherent distortion and response peak those damn AKG things have.
On other forums I've read raving comments about the combo K701 plus Musical Fidelity V8. Never managed to try it, and the V8 isn't a tubes amp, so go figure, but if anything known to man can make those bitches sound like music, it must be that amp