  Theoretical studies of disordered $d$-wave superconductors have focused, with
a few exceptions, on optimally doped models with strong scatterers. Addressing
recent controversies about the nature of the overdoped cuprates, however,
requires studies of the weaker scattering associated with dopant atoms. Here we
study simple models of such systems in the self-consistent Bogoliubov-de Gennes
(BdG) framework, and compare to disorder-averaged results using the
self-consistent-T-matrix-approximation (SCTMA). Despite surprisingly linear in
energy behavior of the low-energy density of states even for quite disordered
systems, the superfluid density in such cases retains a quadratic
low-temperature variation of the penetration depth, unlike other BdG results
reported recently. We trace the discrepancy to smaller effective system size
employed in that work. Overall, the SCTMA performs remarkably well, with the
exception of highly disordered systems with strongly suppressed superfluid
density. We explore this interesting region where gap inhomogeneity dominates
measured superconducting properties, and compare with overdoped cuprates.
