Who said tube headphone amps were the only path? There were lots of hybrid and tube integrated amps since about 1985 with headphone outs, and perhaps directly driven.
This is what I do know: SS amps, even the relative hammer - the Ragnarok 1 in balanced out mode, and the perhaps more audiophile Bryston BHA-1 also in balanced out did not ever approach the musicality of the BHC stock, BHCs (mine well overbuilt), and the XDUOO T-26 - all fairly cheap OTL tube amps do. On paper the Rag 1 delivers 3X more power than the BHC. Now the Violectric V-281 did do better, better than the other two named SS amps - in the bass, but not clear on mids/highs
My idea of music is unamplified music in great halls, and more popular music mixed by Doug Sax, not bass cannons, and stacks of Marshall amps.
Not only do I find this true for the HD-600, but also the HD-800, and HD-800S.
I made the point several times that I did not find tube amps a solution for room speakers. Hybrid pre-amps using small signal tubes not such a problem, but still the issues kept me away until I heard my HD-600 paired with my BHCs probably the cheapest way to get to real musicality (as I see it) avilalbale in either headphones or room speakers. I had the 600 as my only can for 2 years, and one of two for a year on my very functional Ragnarok 1 and never once thought that combo was above the threshold of musicality I get from it the past 3 years or so on the BHCs.
The differences I find are:
* 3 blob sound stage is now 5 blobs. It's not correct, but its better.
* Bass with the BHCs is more robust, but not fat which I do not tolerate.
* harmonics - in particular 2nd, are much closer to real - is that THD? Yeah, but I do find the 600 on SS to be a bit astringent, and that is gone
* timbre due to those changes is better, and the need to quiet some of that low->mid treble stuff is also well mitigated.
I'm just one voice, but, if you check the 800, 800S, and 600 threads here you will find dozens that have arrived at essentially similar conclusions about the sound - as far as how it was developed, probably always an open question.