The constituent quark mass scale is still the same (to within 5-10%) even in what Frank Wilczek calls "QCD Lite" - just two quark flavors with no current mass. So undoubtedly this mass scale is produced within QCD. So far I don't have a simple explanation for its value; we can only hope that there's some simpler way to get it, other than long lattice calculations.
Assuming the connection between the constituent quark mass scale and the Koide relation scale factor is real, it is surely being produced within QCD and transmitted to the leptons. And consider this: simple algebraic transformations of the formula above can bring a factor of 2 out of the squared term, so now we have "mass(lepton) = 2 . mass(constit.quark.) . (new squared term)". In your correspondence, the leptons pair supersymmetrically with mesons, i.e. a quark and an antiquark. So the "naive meson mass", assuming the u/d constituent quark mass scale, is of the order of 2 x 313 MeV.
In other words, one can imagine a sort of "Rivero-correspondence Standard Model Lite", in which all flavors of quark have zero current mass, in which they take on the 313 MeV constituent mass (because of QCD effects) in mesons and baryons, and in which the 625 MeV "naive meson mass scale" gets transmitted to the lepton "superpartners" of the mesons. If such a field theory existed, we could then think about modifying it so that the quarks have nonzero current masses, and so that the charged lepton masses are altered by the extra factor appearing in the Koide formula above.