Encouraged by predictions of simple Fermi liquid behavior in the overdoped versions of high temperature superconductors (HTcSC) based on CuO2 planes, we have performed TF-μ+ SR experiments on several high quality oriented crystal mosaics of Tl2Ba2CuO6+x ("Tl-2201") in the vortex-lattice state where the magnetic penetration depth λ can be extracted from the characteristic superconducting (SC) lineshape by χ2-minimization fitting in the time domain. The results show evidence for a surprising degree of disorder, causing some "smearing" of the characteristic lineshape features, but we were able to extract λ as a function of temperature, revealing a remarkably consistent T-dependence of the SC carrier density ns ∝ λ−2 that suggests this "role-model cuprate" is anything but "simple". The extracted superfluid density for all dopings exhibits a clear inflection point with temperature, suggesting a vortex lattice transition. Such a transition would have important implications for other measurements.