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The Kondo effect is an archetypical phenomenon in the physics of strongly correlated electron systems. Recent attention has focused on the application of Kondo physics to quantum information science by exploiting overscreened Kondo impurities with residual anyon-like impurity entropy. While this physics was proposed in the fine-tuned multi-channel Kondo setup or in the Majorana-based topological Kondo effect, we here study the Kondo effect with symplectic symmetry Sp(2k) and present details about the implementation which importantly only involves conventional s-wave superconductivity coupled to an array of resonant levels and neither requires perfect channel symmetry nor Majorana fermions. We carefully discuss the role of perturbations and show that a global Zeeman drives the system to a 2-channel SU(k) fixed point. Exact results for the residual entropy, specific heat, and magnetization are derived using the thermodynamic Bethe Ansatz for Sp(2k). This solution not only proves the existence of a quantum critical ground state with anyon-like Hilbert space dimension but also a particularly weak non-Fermi liquid behavior at criticality. We interpret the weakness of non-analyticities as a manifestation of suppressed density of states at the impurity causing only a very weak connection of putative anyons and conduction electrons. Given this weak connection, the simplicity of the design, and the stability of the effect, we conjecture that the symplectic Kondo effect may be particularly suitable for quantum information applications.