Despite governing heat management in any realistic device, the microscopic mechanisms of heat transport in all-solid-state electrolytes are poorly known: existing calculations, all based on simplistic semi-empirical models, are unreliable for superionic conductors and largely overestimate their thermal conductivity. In this work, we deploy a combination of state-of-the-art methods to calculate the thermal conductivity of a prototypical Li-ion conductor, the Li₃ClO antiperovskite. By leveraging ab initio, machine learning, and forcefield descriptions of interatomic forces, we are able to reveal the massive role of anharmonic interactions and diffusive defects on the thermal conductivity and its temperature dependence, and to eventually embed their effects into a simple rationale which is likely applicable to a wide class of ionic conductors. In this record, we provide data and scripts to generate the plots supporting our findings. We also provide the machine learning model and the dataset to train it.