My MK I unit has no relais, either, but has the 5V heaters, so that's probably unrelated.
As I said, rising the heater voltage to 5.8V didn't make a huge difference. It certainly didn't cure all that's wrong with this model.
I think underheating a tube mic and underheating a preamp/channel strip are not the same thing, as the latter has to deal with higher signal levels at least in the output stage. Also, 5.8V is about the voltage Gefell run their tube mics, but not 5V.
If you've read Pip's comment: he has a Solo 610 and an LA-610, and he confirms my observations: The Solo 610 sounds good, the LA-610 doesn't. Actually, I bought the LA-610, because I thought it would sound as good as the Solo 610. Well, it doesn't. (BTW I liked the Solo 110 even better)
So I don't think it's a philosophical question. I'm not a newbie, either, I've been recording for 20+ years, and I own some (real) vintage stuff, and I know how to use my gear, new and old. Granted, you can make the LA-610 sound real nasty if you don't know its workings, but you can't make it sound really good no matter how you use it.
One more thing: The distortion characteristics don't change much with the gain switch. In my measurements the distortion spectrum remained basically the same. You can manipulate the distortion level (not the spectral distribution!) a bit by using less input gain and more output gain, but not much. Using an additional gain stage behind the LA-610 is a bit more effective. In both cases you lose some S/N, obviously.