So, what's the problem, beyond the obvious issues of computing efficiency and the fact that we don't know how to write down an exact form for the exchange-correlation part of the functional (basically where all the bodies are buried)?
Well, the noninteracting states that people like to use, the so-called Kohn-Sham orbitals, are seductive. It's easy to think of them as if they are "real", meaning that it's very tempting to start using them to think about excited states and where the electrons "really" live in those states, even though technically there is no a priori reason that they should be valid except as a tool to find the ground state density. This is discussed a bit in the comments here. This isn't a completely crazy idea, in the sense that the Kohn-Sham states usually have the right symmetries and in molecules tend to agree well with chemistry ideas about where reactions tend to occur, etc. However, there are no guarantees.
There are many approaches to do better (e.g., some statements that can be made about the lowest unoccupied orbital that let you determine not just the ground state energy but get a quantitative estimate of the gap to the lowest electronic excited state, and that has enabled very good computations of energy gaps in molecules and solids; time-dependent DFT, which looks at the general time-dependent electron density). However, you have to be very careful. Perhaps commenters will have some insights here.
The bottom line: DFT is intellectually deep, a boon to many practical calculations when implemented correctly, and so good at many things that the temptation is to treat it like a black box (especially as there are more and more simple-to-use commercial implementations) and assume it's good at everything. It remains an impressive achievement with huge scientific impact, and unless there are major advances in other computational approaches, DFT and its relatives are likely the best bet for achieving the long-desired ability to do "materials by design".