DFT is based on two important theorems:
(1) Hohenberg & Kohn: the potential and the density are connected by a one-to-one map
(2) Kohn & Sham: there is always a non-interacting reference system (map: V_xc: non-interacting <-> interacting problem) having the same density as the interacting one.
In a nutshell: the potential and the density of the interacting system can be represented by a non-interacting potential / density.
So, DFT itself is exact in the ground state charge density if one knows the exact V_xc. Usually, V_xc is taken for a system where we have access to both solutions: the interacting and the non-interacting one. The most common reference system is the homogeneous (non-)interacting electron gas.
To your question: strictly speaking, transport properties are excitation properties. Thus engineer is correct in that point. The Kohn-Sham eigenvalues are the eigenspectrum of the non-interacting reference system and not the spectrum of the interacting problem (they might be totally different)! Surprisingly, it turned out that the Kohn-Sham spectrum is for many cases close to the excitation spectrum. The interpretation, however, as an excitation spectrum is mathematically not justified. It is only valid for Hartree-Fock (see Koopman's theorem). So the whole business of "predicting" band gaps within DFT(optimized V_xc) is emperically founded.
A comment to PuZhang: of course, one can improve V_xc's, but in order to interpret the Kohn-Sham eigenstates as excitations, and thus to derive "band gaps", one has to proceed in a different way. During the derivation of the Kohn-Sham equations, one can add a constraint forcing the eigenvalue spectra to be identical between the interacting and non-interacting system. However, whether one is still capable of finding a suitable approximation to V_xc in that case is yet to be proven.
All the best,