Try listening at reduced volume, and I think you will find that triode has a decided advantage over tetrode, even with large-scale, complex orchestrations; the issue is simply that to unravel and amplify a complex signal, more power is needed. It's not that tetrode ever has an advantage, per se, it's that when the music is more demanding, the extra power makes up for any loss in transparency that triode provides under ideal circumstances (higher speaker efficiency). Conversely, try turning the volume up, higher than normal, while playing small cale jazz or classical. I think you will find that you lose the detail and purity.
I think that we can all agree that filet mignon is a better cut of beef than sirloin; but if the budget only allows for one pound of filet, and we are trying to feed six adults, would it not be more satisfying to buy two pounds of sirloin, than one pound of fillet. Poor analogy, perhaps, but the point is that I think that if you were to compare tetrode to triode, using something along the lines of a Klipsh corner horn with it's extraordinarily high efficiency, it's unlikely that there would be any advantage to tetrode mode; unless, perhaps, if you were trying to fill some incredibly large space with sound.
By the way, it is not necessary to turn the amps off to switch between the two modes. You simply have to be sure there is no signal running through them: mute the preamp.