A warm DAC?

Chord qutest. Analog sounding that presents the music accurately and uncolored. Just might fit your system. 

What is your budget $3k-5-$$10k ?

As I stated earlier, I like Audio Note's philosophy; specs don't matter as long as it sounds good and it's faithful to the original recording.

Ideally a DAC is designed to be flat across the frequency range of our hearing, so any deviations from the ideal are considered non-linearities aka distortion. If the DAC adds distortion how can it be faithful to the original recording? In vinyl this is a given there is distortion, but in digital any noise or distorion in well designed DACs have been pushed beyond human hearing. To purposely introduce distortion is not being faithful to the original recording. 

I’ll agree with your last statement. Although the distortion may be a manufacturing flaw and unintentional. Or not audible.
But I still ask, how does it sound?

The Audio Mirror Tubadour III SE is an excellent DAC.  I honestly preferred it over a DirectStream Sr.  I'm using it with a Herron pre amp and Innous Zenith MKII streamer.