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Several recent experiments have challenged the premise that cuprate high-temperature superconductors approach conventional Landau-BCS behavior in the high-doping limit. We argue, based on an analysis of their superconducting spectra, that anomalous properties seen in the most-studied overdoped cuprates require a pairing interaction that is strongly inhomogeneous on nm length scales. This is consistent with recent proposals that the “strange-metal” phase above $T_c$in the same doping range arises from a spatially random interaction. We show, via mean-field Bogoliubov-de Gennes (BdG) calculations and time-dependent Ginzburg-Landau (TDGL) simulations, that key features of the observed tunneling spectra are reproduced when both inhomogeneity and thermal phase fluctuations are accounted for. In accord with experiments, BdG calculations find that low- $T$spectra are highly inhomogeneous and exhibit a low-energy spectral shoulder and broad coherence peaks. However, the spectral gap in this approach becomes homogeneous at high $T$, in contrast to experiments. This is resolved when thermal fluctuations are included within TDGL; in this case, global phase coherence is lost at the superconducting $T_c$via a broadened BKT transition, while robust phase-coherent superconducting islands persist well above $T_c$. The local spectrum remains inhomogeneous at $T_c$, and the gap is found to fill instead of close with increasing temperature.