I really think the lack of appreciation for the S models boils down to the more expensive, excellent build, which is difficult to communicate through the internet-direct sales model, and then there is also the positioning right along side the incredibly novel + modes which have the tube mode you can switch on and off. That adds a lot of value for those who are curious about tubes. The + models are really pretty too.
Looking at the tube specs
Schiit site, Saga+:
THD: <0.001%, 20Hz-20KHz, at 1V RMS
IMD: <0.001%, CCIR (active stage)
SNR: >108db, A-weighted, referenced to 1V RMS
Freya+:
THD: <0.01%, 20Hz-20KHz, at 1V RMS
IMD: <0.01%, CCIR
SNR: >115db, A-weighted, referenced to 2V RMS
The circuits are very well implemented and it is not surprising you don't hear the tubes. A rougher implementation, like a Darkvoice 336SE, or maybe one of the really cheap Chinese tube buffers or HPAs with buffer stage pass through might give you the sound you are looking for. Stay away from the Loxjie P20. I haven't see specs yet, but my subjective experience, and that of others, is that the sound is very clean. I love the P20, but but it is not my recommendation for adding audible distortion and frequency response change.
Maybe
this guy. One review on Amazon (David Roehr), has some AP measurements showing lots of harmonic distortion and modified FR.