$\gamma$-$B{a}_{3}CoN{b}_2{O}_9$ realizes a disordered simple-cubic spin-$1/2$ lattice in which Co${}^{2+}$ ions randomly occupy one-third of the sites, placing the system just above the site-percolation threshold for magnetic order. Specific-heat, susceptibility, neutron spin-echo, and muon spin-rotation measurements reveal a broad thermodynamic crossover, short-range magnetic correlations, and persistent fast spin dynamics down to at least 0.1 K, with no evidence for static order or conventional spin-glass freezing within the experimental time windows. Monte Carlo simulations yield a broad distribution of orphan spins, finite clusters, and an infinite network. The weakly correlated spin fraction ($\approx 8.2\%$) inferred phenomenologically from magnetization is consistent with the calculated orphan-spin fraction ($\approx 8.8\%$). Exact diagonalization of a diluted $S=1/2$ Heisenberg model captures the broad magnetic specific-heat anomaly and supports the coexistence of weakly and strongly correlated spin environments. These results support a picture in which spin-$1/2$ quantum fluctuations, together with dilution and proximity to the percolation threshold, can support a disorder-driven dynamical state with short-range correlations in three dimensions, distinct from both classical spin glasses and geometrically frustrated quantum spin liquids.