What is “warmth” and how do you get it?

Play "Light My Fire" over and over again.

I agree with Hifiman. Your system has to have enough 'warmth' inorder to portray music naturally and realistically. I think it's a matter of degrees: too little warmth and the result will be overly sterile and analytical, too much warmth and you get a syrupy sound (which I define as slow with an emphasis in the upper bass/lower midrange). I think of it as a continuum:

analytical___________________________syrupywarm

The trick is to first determine generally where on this continuum your taste lies and then,secondly, to assemble your system components to match your tastes. Easier said then done, for sure. I would say most A'gon members' tastes lie between somewhere, where the warmth is, with very few at the extremes.
Looking back on my system evolution, I clearly moved from an original analytical system to a current system that is much warmer without consciously thinking in these terms. But I can remember that I was looking for more realism and that illusion of 'you are there'. I know now that I was wanting more 'warmth',depth and presemce.
But your question was about how do you get more warmth. There was another recent thread about 'bloom' that you may want to read. I think the 2 concepts are closely related. In my limited experience, I would suggest tubed equipment needs to exist somewhere in your audio chain (source, preamp,amp) to obtain good, natural warmth and bloom. I don't deny you can achieve some warmth and bloom with solid state equipment but you'll need to select carefully and have deep pockets to be successful. Speaking of which, your system is very nice and I don't see any obvious cause of a lack of warmth. I don't think your Focal spkrs or Pass amp are responsible. I am definitely no expert but that won't stop me from making one suggestion: experiment. You could try out a few tubed preamps to see if this gives you the desired warmth(hopefully, there are some local dealers that will accomodate this). This worked for me. I'm not familiar with your preamp/processor or the mods but know you've made a big investment in it. But even if you substituted a different preamp, you could still utilize your current preamp as a prossessor if the new preamp has a HT passthru. Not sure if I helped but wish you good luck on your search.

According to Technical Director of Benchmark John Siau, DAC1 that I own, was specifically designed not to sound warm but rather natural without enhancement of any harmonics odd or even. Enhancing even harmonics does sound really nice with guitar or voice but not so with instruments that poses a little more complex than regular overtones harmonic structure. Piano is such instrument (as well as percussion instruments) and when it is reproduced on very warm equipment it sounds almost like out of tune.

On the other hand SS gear often designed to sound hi-fiish or for good spects on paper enhances (because of transient intermodulation introduced by deep negative feedback) odd harmonics. Our ears are very sensitive to odd harmonics since loudness cues reside there (like 7th or 9th harmonic).

Neutral sound (whatever it is) is the best but I understand that staying a little on warm side protects from brightly recorded material. Fortunately my Hyperion speakers sound slightly warm with zero sibilants on any CD - and it is in combination with Benchmark DAC1 and unforgiving class D amp.
My older speaker had aluminum dome tweeter and was unbearably bright in the same setup.

Your post should have read , " Warmth why would you want it ". Warmth is an added artifact of electronically produced music . A properly tuned acoustical instrument should not have the added warmth that you here on many systems . Having tuned musical instruments in the past , one thing we listened for was any unnatural warmth , if you could hear warmth something was wrong . Warmth was never considered natural or desirable . It would altar pace, pitch, flow and timing and make an instrument sound less neutral , most easily noticeable on sting instruments . Have A musical day

Having an always "warm" system is like a preference, choice , or flavor and may not be for everyone. I have a small body acoustic guitar and if played back through my system I wouldn't want or expect it to sound warm. I also have a big bodied mellow and bassy sounding guitar and wouldn't want that one to sound thin or bright. I can choose which guitar to play for different sound but I want my system to be neutral and have the capability of being true to the source. That's my preference.