Recently, we have argued that experimental data on superfluid density and terahertz conductivity of overdoped LSCO are compatible with a Landau Fermi liquid/Bardeen-Cooper-Schrieffer description of these samples, provided dopants are treated within dirty $d$-wave theory as weak scatterers. Here we test these ideas by comparing to specific heat and thermal conductivity data on LSCO, showing that the theory works extremely well across the overdoped region for similar disorder parameters. We then study the same properties in another overdoped cuprate, Tl-2201, thought to be quite clean since it exhibits quantum oscillations, low residual resistivities and small superconducting state Sommerfeld coefficients. Our results are consistent with the Tl-2201 system being $approx 3$ times cleaner due in part to the dopant atoms being located further from the CuO$_2$ plane. We conclude that cuprates can be described semiquantitatively in the overdoped regime by dirty $d$-wave theory, subject to significant Fermi liquid renormalizations, without introducing physics beyond the Landau-BCS paradigm.
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