I think of warmer sounding equipment as being closer to hearing sounds naturally. I think sound has a warm profile. Eg bird-song is not cold or clinical like some audio equipment. I think most sound has a warm texture. (I guess that's why sometimes you can fall asleep listening to sounds in a warm bubble.)
When listening to bird song, I agree it is not cold and clinical, but then neither is any neutral sound
However, I don't have many nature albums. But I do listen to quite a few bands up close in local venues. When I am near a drummer, a struck cymbal sounds quite clinical and many would say harsh, but that is how it sounds. I like to get this sound when I hear a live or studio recording, but I feel so much equipment now rolls off this impact, creating a warm cosy sound. Maybe this is to appeal to people who have grown up on mp3s, streaming, bass heavy headphones from you-know-who and generally an intolerance to crisp treble of the digital age.
I heard the respected Oppo PM3 headphones and couldn't believe the warm cosy sound with so little impact. They had a lovely sweet midrange, but overall not for me, if I hear a rock band, they got to rock! Great for those who want that type of lush sound.
I thank Chord for producing some of the most natural, neutral equipment available, starting at the bargain price Mojo, the amazing Hugo, and the fantastic sounding TT. I feel only the Mojo has the slightest bit of warmth, if a few people find that clinical, they are probably looking at the wrong brand.