After listening to about HD600 for about 30 minutes with NFB-11, I can definitely say that the match is better than HD650 for people with sound signature towards neutral/treble. I am not saying that NFB-11 is horrible with HD650, but it sounded a little veiled and too warm/mellow for me. That's just my case, people have different tastes and if you like smooth, laid back sound, go for the HD650...otherwise, I think HD600 is a nice match.
While I think soundstage and bass was a bit better on the HD650, but HD600 is not too much different. The biggest difference is the treble and the laidback sound of HD650, other than that I think they are pretty similar. I have very recent productions of both, so it might be different if you have much older pairs.
You won't be let down with HD600+NFB-11
If you are having trouble with wasapi in foobar2k about the mute relay, use post/pre-track silence component to set affix silence DSP (2000 milliseconds of silence before is doing well for me).
Also, I don't think anyone will disagree to the fact that HD650 has much better stock cable...I think build quality of HD650 was better...