We are interested in establishing weak and strong well-posedness for McKean-Vlasov SDEs with additive stable noise and a convolution type non-linear drift with singular interaction kernel in the framework of Lebesgue-Besov spaces. In particular, we characterize quantitatively how the non-linearity allows to go beyond the thresholds obtained for linear SDEs with singular interaction kernels. We prove that the thresholds deriving from the scaling of the noise can be achieved and that the corresponding SDE can be understood in the classical sense. We also specifically characterize in function of the stability index of the driving noise and the parameters of the drift when the dichotomy between weak and strong uniqueness occurs.