A coarse-grained model of the action of a chaperonin cage of tunable hydrophobicity, h, upon a protein with the possibility of misfolding is studied with inherent structure (IS) analysis and statistical temperature molecular dynamics (STMD) simulation. Near the folding temperature, the equilibrium properties of the system may be understood in terms of <10 IS. The known phenomenon of an optimal cage hydrophobicity for productive folding, found at h = 0.25, is seen to arise from a striking suppression of the occupations of IS in the misfolding funnel, which in turn arises from a decrease in translational entropy due to confinement to the region of the cage wall. The kinetics of folding is correspondingly fastest at h = 0.25, where a minimum is found in the h-dependent barrier height. While true kinetics is determined by conventional MD, it is shown that the accelerated dynamics of STMD provide a valuable quantitative perspective.