Laves-phase Nb(1+c)Fe(2-c) is a rare itinerant intermetallic compound exhibiting magnetic quantum criticality at c(cr)∼1.5%Nb excess; its origin, and how alloying mediates it, remains an enigma. For NbFe(2), we show that an unconventional band critical point above the Fermi level E(F) explains most observations and that chemical alloying mediates access to this unconventional band critical point by an increase in E(F) with decreasing electrons (increasing %Nb), counter to rigid-band concepts. We calculate that E(F) enters the unconventional band critical point region for c(cr) > 1.5%Nb and by 1.74%Nb there is no Nb site-occupation preference between symmetry-distinct Fe sites, i.e., no electron-hopping disorder, making resistivity near constant as observed. At larger Nb (Fe) excess, the ferromagnetic Stoner criterion is satisfied.