This is true, there are even "comma pump" sequences that are literally impossible to play fully harmonically, being inherently based on conflating "close" ratios! (I think in theory, a very detailed Schenkerian-ish analysis of a piece would tell you quite a bit about where ratios should be made "nicer" and where they shouldn't - but that's a lot of work and involves a whole lot of personal judgment.)
I think one of the reasons ii V I is so compelling (versus IV V I) is that the ii (or ii7) is conflating many notes. In C major, using interval terms as a stand-in for "tuned justly":
Is the D a fourth below G or a minor (or harmonic) seventh below C? Once you've chosen your D, is A a perfect fifth above it? If so, what kind of major second relation does it now have to G?
Is the F a perfect fourth above C, or is it a major third above your D? Or is it supposed to be the minor (or harmonic) seventh above G?
I think barbershop tends to skirt these questions by instead using a II V I progression (a secondary dominant), since the F# doesn't have to stand in careful relation to other notes of the plain C major scale.
(That's interesting Schenkerian analysis can help understand these issues. I know only about it at a very thin surface level.)
I suppose a good rule of thumb is that "dissonant" notes (in the musical syntax sense) should be in a harmonic relation with the scale step they will ultimately resolve into - so D to C, A to G, F to C etc. This can and will create roughness in the vertical dimension, but that actually reflects their unstable "energy" that drives them to resolution. Of course one could just as well use microtonal shifts to play with this kind of stability whenever longer-range relations are involved. Perhaps this is part of what good performers do intuitively when playing a melodic instrument. The relevance of music-syntactical dissonance and resolution is what might make Schenkerian analysis a very useful tool to understand these possibilities!
