The search for novel materials capable of robust spin-current generation remains an active area of research. Here, we investigate the intrinsic spin Hall and spin Nernst effects in Co- and Rh-based skutterudites and explore their enhancement through heavy-metal substitution and site-selective doping. Using first-principles calculations and Wannier-based tight-binding models, we analyze the spin Berry curvature and spin-transport responses of pristine MX₃, where M = Co, Rh and X = As, Sb, as well as chemically modified systems. Despite sizable momentum-resolved spin Berry curvature, the pristine compounds exhibit relatively small net spin Hall responses due to substantial cancellation across the Brillouin zone. Isoelectronic Ir and Bi substitution at 8c Wycoff M site produces little enhancement near the E_F, whereas non-isoelectronic Pt substitution yields a relatively better response. Most interestingly, the incorporation of Pt at the interstitial 2a Wycoff site of RhAs₃ nearly doubles the spin Hall conductivity to ∼400 (ℏ/e)(Ω,cm)⁻¹ near the E_F when compared to Pt substitution at 8c position. At 300 K, the same interstitially filled system exhibits a spin Nernst conductivity of ∼1.46 (ℏ/e),A,m⁻¹K⁻¹ at the E_F, nearly an order of magnitude larger than that of M-site Pt substitution. The enhanced responses arise from Pt 5d states, strong spin-orbit coupling, and avoided crossings that generate pronounced spin-Berry-curvature hot spots. These results demonstrate that site-selective chemical engineering can effectively overcome momentum-space cancellation resulting in enhanced spin Hall and spin Nernst responses in skutterudites. Our results provide a possible route towards robust spin-current generation in this class of materials.