The density functional based tight-binding (DFTB) method can benefit substantially from a number of developments in density functional theory (DFT) while also providing a simple analytical proving ground for new extensions. This contribution begins by demonstrating the variational nature of charge-self-consistent DFTB (SCC-DFTB), proving the presence of a defined ground-state in this class of methods. Because the ground state of the SCC-DFTB method itself can be qualitatively incorrect for some systems, suitable forms of the recent LDA+U functionals for SCC-DFTB are also presented. This leads to both a new semilocal self-interaction correction scheme and a new physical argument for the choice of parameters in the LDA+U method. The locality of these corrections can only partly repair the HOMO-LUMO gap and chemical potential discontinuity, hence a novel method for introducing this further physics into the method is also presented, leading to exact derivative discontinuities in this theory at low computational cost. The prototypical system NiO is used as an illustration for these developments.