There is a hot debate on the anomalous behavior of superfluid density $rho_s$ in overdoped La$_{2-x}$Sr$_x$CuO$_4$ films in recent years. Its linear temperature dependence $rho_s(0)-rho_s(T)propto T$ infers the superconductors are clean, but the zero temperature value $rho_s(0)propto T_c$ is a hallmark of the dirty limit in the Bardeen-Cooper-Schrieffer (BCS) framework (Bozovic et al., 2016). In this work, we show that the apical oxygen vacancies can lead to an anisotropic scattering rate $Gamma_dcos^2(2theta)$, which can explain the above two linear scalings simultaneously, and thus provides a plausible solution to this clean-dirty paradox. Furthermore, by analyzing the optical conductivity, it may also explain the ``missing Drude weight upon doping as reported in the THz experiment (Mahmood et al., 2019). Therefore, we conclude that the superconducting states of the overdoped cuprates are consistent with the disordered BCS theory.
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