The increased capabilities of coupling more and more materials through functional interfaces are paving the way to a series of exciting experiments and extremely advanced devices. Here we focus on the capability of magnetically inhomogeneous superconductor/ferromagnet (S/F) interfaces to generate spin-polarized triplet pairs. We build on previous achievements on spin-filter ferromagnetic Josephson junctions (JJs) and find unique correspondence between neat experimental benchmarks in the temperature behavior of the critical current and theoretical modeling based on microscopic calculations, which allow us to determine a posteriori spin-singlet and triplet correlation functions. This kind of combined analysis provides accurate proof of the coexistence and tunability of singlet and triplet transport. This turns to be a unique opportunity to model disorder and spin-mixing effects in a JJ to enlarge the space of parameters, which regulate the phenomenology of the Josephson effect and could be applied to a variety of novel types of JJs.