Good to know!
Ok, I got it sorted, thanks to @Dsmnoisemaker. There is no defect, just some lack of understanding, and a bit of user error.
Daniel wrote this to me: "the aux is mixed with the signal via resistors, so when the aux is connected, there's a 6dB drop on the bass signal. It's normal." After reading that, I experimented a bit more. I realized that really what I was hearing was not "signal degradation", but a simple volume drop. Short fix: just bump the headphone volume a bit, and the same sound is there. No need to touch gain or eq. Very easy! Honestly, I feel a bit stupid.
As for the preamp loop thing: well, the issue could be my overdrive pedals, or again some kind of user error. Anyway, I don't think I will be using that any time soon, and if I do, I will experiment more.
So everything is great! This box has quite a learning curve (at least for me), but it is truly amazing. Great sound shaping on its own, as well as after overdrives. Pretty much allows me to live without an amp (I move around a lot, so I need to keep things light). Together with some recent additions of high-quality cables and a quiet power supply, everything sounds fantastic.
Thank you DSM and Humboldt!