Hi, I own some of the headphones mentioned here and some others, all below $500. What I'm looking for in a headphone is mostly neutral sound with naturally sounding midrange and treble (no grittiness, sharpness etc in vocals and cymbal sounds, no sibilance, general smoothness with detail intact) with slight bass boost, but nothing overpowering. Short version is that after owning Beyer DT880 and 990 (600 ohm), DT770 (600 and 80 ohm), Senn HD650, 58X and Momentum 3 (and others), Fidelio X2HR, Denon AH-D2000 and AH-D5200 and E-MU Teak and Walnut, the most happy I am with is the E-MU Teak. It sounds like it's a level above all the others, especially when it comes to dynamics and "punchiness" of the sound. Some of my friends who listened to them disliked them for having "too much bass", which might be the case, they do sound like they have the most bass from all the HP's in my collection (except maybe for the Momentum 3), I like it though. Most people who heard them, however, loved the sound. I love their sound.
If I had to recommend "neutral" headphones to someone, I would recommend the Sennheiser HD 650 or 58X (or 6XX - same as 650 but cheaper, only on Drop, 58X is also Drop-exclusive). The 650's might be a bit more "smooth" and "natural" in treble than the 58X but generally I think I would have trouble telling them apart in a blind test.
I would definitely not recommend the DT880 (or any of the Beyers I've had - DT990 or 770) - the Beyers simply have too much high frequencies to be neutral, they sound thin and sharp and "crispy". If you don't mind slightly added bass, then I would recommend the Teaks or AH-D5200. The Denons have less dynamic sound, more upper midrange/lower treble and less prominent bass. The Teaks sound more smooth. However I think they are not exactly built for studio, they might break if you don't treat them carefully. The Denons feel way more sturdy.
I haven't heard the 560S, but have heard good things about them, but they seem to have an ugly peak in upper midrange, which might make them boost the "presence" area in vocals and electric guitar, for example. The K371 look generally very good, from what I've read about them. Haven't hear them neither, though.
Crinacle measured most of the headphones mentioned here, including your R70x, so you can have a look at his graphs, compare the headphones' frequency response and try to decide based on those.
https://crinacle.com/graphs/headphones/graphtool/