Recommend A New DAC Between $1000 and $2000

For your budget and checklist: Denafrips Pontus II near the top of the range and Ares II just over the bottom of the range. Warm rich and will show you why R2R DACS rule.

I'd recommend a RME ADI-2 FS as a balanced DAC with a great remote.  This DAC preamp is unique in its features and ability to move the sound shape and to do parametric EQ digitally upstream of the analog signal.  You can convert your headphones or speakers to basically anything you would like it to be.  The remote has very fine incremental L/R Balance, Bass and Treble control.  You also can control the parametric EQ individually on the Left or Right Channel - great for hearing loss in one ear or EQing individual speakers in a room to compensate for room placement.  Or just use it like a great DAC on pure line output or a high power single-ended or very low noise IEM DAC/AMP combo.  Providing a DAC with variable volume control (with any EQ)  to a pair of powered speakers, it is impossible to beat in my opinion.

I second the Rme adi2 suggestion. It is a gem.

i had an rme adi2 about two years ago, the fs2 unit with akm chip

i felt it was a feature rich, well built, very clear and precise sounding dac, best run without internal volume attenuation (as this was done in the digital domain and could reduce resolution if high attenuation levels are used) - a great value in terms of sound and control over the sound, for the money...

at the time i also had the ayre codex, mhdt stockholm and chord qutest in house for comparison, and i felt the rme was excellent in its quite precise, very clear presentation of the music with extended frequency response -- but its weaknesses were a more mechanical, less organic presentation than either the ayre or mhdt (a nos tube buffered dac), and had a notable stronger central image without the appreciable soundstaging width or depth of the other dacs (the chord was the best in this regard, had tremendous spatial imagery, followed closely by the mhdt then the ayre) -- as such i concluded the rme is best suited for headphone listeners where imaging ability is less valued than the other positive attributes the rme brings its user