To clarify the mechanism of the zero-tuning quantum criticality as well as the robust criticality, we analyze theoretically electronic states of quasicrystal Yb₁₅Al₃₄Au₅₁ and its crystalline approximant [3]. By constructing a minimal model for both systems, which contain concentric shell structures with Yb and Al-Au clusters, we show that quantum critical points of the first-order valence transition of Yb appear as spots in the ground-state phase diagram with their critical regimes being overlapped to be unified, giving rise to a wide quantum critical regime. This well explains the robust unconventional criticality observed in Yb₁₅Al₃₄Au₅₁ under pressure. The formation of the wide critical-regime reveals that zero-tuning quantum criticality is not caused accidentally. Our results also predict that the common unconventional criticality will also appear in the crystalline approximant when pressure is tuned [3,4]. The key origin of these striking phenomena is ascribed to the atomic origin of the valence transition and consequent locality of the critical valence-fluctuation mode. Regardless of periodicity or quasi-periodicity of lattice arrangement, new universality class is considered to be formed by critical Yb-valence fluctuations.