I think you are mixing up things here a bit. First of all, without periodic boundary conditions there are no bands, but still electronic states which as you correctly say are effective single particle states. In general, these states have to be taken with care, since they have been derived from a single particle electron density representation, i.e.
$\rho = \sum\limits_l f_l|\psi_l|^2$
where $f_l$ is the occupation of the states and the $\psi_l$'s are the single electron wave functions. The physical meaning of the resulting states is mostly unclear. It has been however shown that the HOMO and LUMU correspond to vertical ionization potential and vertical electron affinity. From this fact, one can derive the HOMO-LUMU gap.
Under the influence of periodic boundary conditions, each state is broadened into bands. The HOMO-LUMU gap becomes the band gap.
The HOMO-LUMU gas is indeed the actual band gap, under the consideration of electronic entropy which smears out the bands . There are of course systems where certain functional are bad, but this is not a fault of DFT, but it's parametrization. For example semiconductors are intrinsically difficult to describe on the GGA level due to electron correlation having a huge influence on the band gap and GGA's artificially smearing out the electron density.
Concerning excited states, DFT can be modified by approximating excitation energies (e.g. Delta SCF) or extended to TDDFT for a more exact answer.