There's really no magic in DACs. They convert digital to analog as precisely as possible.
Sound stage is mainly a property of the recording, but influenced to some degree by speakers/room or in your case headphones and ear geometry. NOS is technically not a filter at all and is
by far the worst and most imprecise way to convert signals into the analog domain. It generates tons of ultrasonic noise and it shows a light roll-off in the treble region.
Your
DT1990s are quite hot above 6 or 7 kHz, which is well known to create a divide in it's user base: Some love it, others hate it for that quality. It seems you don't actually like the overly hot treble and NOS mode lowering the power in that region makes them listenable to you.
My recommendation: Skip NOS mode, use a regular fast oversampling filter and dial in some EQ to tame the treble. It's much more flexible and precise than the "take it or leave it" approach of using a broken thing like NOS.
There's nothing special sound wise about R2R DACs. They often come with a NOS option, which is why many people tend to mix up these two properties. In essence, you can build audibly transparent R2R DACs and the same is true for delta-sigma designs. R2R will on average have a slight disadvantage in measured performance, because the tech is simply inferior.
Not sure if the K11 offers EQ, but the K13 should as far as I can see. That's a definite advantage in your case and can help you make the DT1990 more agreeable.