I kind of understand what you're saying.
I don't have HD560S but I do have HD600 and HD58X. The Sennheisers have the more strictly correct midrange, vocals are more properly rounded and tonally correct - particularly with female vocal, less so on male vocal. I hear the same thing if comparing the K712 to my ATC speakers. Curiously I don't find this bothersome with the K712.
The HD600 and K712 have similar sensitivity and I have done lots of swapping between these two headphones. Without comparing directly to the HD600, vocals sound quite good on the K712. The K712 has superior articulation of instrumental detail - hearing instrumental lines, playing techniques, separation of instruments and just generally whats going on in the mix - to me this makes music more interesting and engaging.
With the K712 I want to listen to it for longer. The HD600 sounds a bit staid by comparison. Though I respect the HD600 as a fine headphone for the money, I seldom listen to it for long sessions. I end up wanting to switch to something else.
Part of this may be that I find the perspective and presentation of the K712 more speaker like, less forward and confined in the head than the HD600.
Both are fine headphones for not a lot of money but, for enjoyment and involvement listening to music of all genres, I would choose the K712 over the HD600 in a heartbeat. Maybe it's an acquired taste?
I honestly kept letting go of the HD 600/650 numerous times because they fail to engage me and I rarely reached out to listen to them. But they have fantastic tuning in their midrange and I respect what they do well. Honestly the musical engagement of the K712 is much higher to me than the HD 560 S, HD 600, and HD 650. I did get a lot of engagement out of the HD 660 S though. But the HD 660 S fatigued me some reason, which upset me as it was what I wanted the HD 600/650 to be in terms of engagement and sound.
The K712 does have some midrange coloration going on and it’s cheaper sibling the K612 is less colored, more neutral, and honestly clearer sounding but it lacks some of the technicalities and finesse of the K712. Despite the K612 being more neutral I prefer the K712. On the HD 560 S I had a love/hate relationship with them, they fixed some of the roughness the HD 600/650 have in their upper midrange and they had nice bass extension and their tuning was good if a bit bright; but they lack body, depth, and detail in to their sound in comparison, they don’t scale as well, and their soundstage is quite narrow to my ears (which I don’t mind but it felt a bit artificially constrained some reason). It felt like the HD 560 S was a headphone that had potential but was held back somehow.