  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. The linear
drop of $\rho_s$ at low temperatures implies the superconductors are clean, but
the linear scaling between $\rho_s$ (in the zero temperature limit) and the
transition temperature $T_c$ is a hallmark of the dirty limit in the
Bardeen-Cooper-Schrieffer (BCS) framework [I. Bozovic et al., Nature 536, 309
(2016)]. This dichotomy motivated exotic theories beyond the standard BCS
theory. We show, however, that such a dichotomy can be reconciled naturally by
the role of increasing anisotropic scattering caused by the apical oxygen
vacancies. Furthermore, the anisotropic scattering also explains the "missing"
Drude weight upon doping in the optical conductivity, as reported in the THz
experiment [F. Mahmood et al., Phys. Rev. Lett. 122, 027003 (2019)]. Therefore,
the overdoped cuprates can actually be described consistently by the $d$-wave
BCS theory with the unique anisotropic scattering.
