When we renormalize to obtain the physical mass, the $Lambda$ dependence of the physical mass is removed by introducing the counterterms in the Lagrangian. So whether we put $Lambdarightarrowinfty$ or not, it does not matter. Even we can take $Lambdarightarrowinfty$ at the end of the calculation. But can we really take $Lambdarightarrowinfty$ limit (even at the end of the calculation) if a quantum field theory is not complete at high energies? If yes, what would that mean? Certainly it cannot imply that since we can take $Lambdarightarrowinfty$, the QED is valid at all energies. Because we know it is not. I think that this point is blurred in all QFT texts.
Moreover, since the $Lambda$ dependence of the physical mass is removed due to the counterterms, doesn’t this imply the physical mass (of the electron, say) has a constant value, independent of the cut-off?