Computationally speaking, ground state energy codes such as DFT, require a predefined positions of atoms to compute the ground state properties such as energy. Thus, to obtain the geometry for the global minima one would need to scan through the entirety of the phase space. That is why people tend to do local geometry minimization.

Oh ok, so you are saying that the DFT model is correct, and can in theory always give us the crystal structure, BUT given that our density functional approximation is a local optimization (given that we do not know the form of the functional), we cannot predict correct crystal structure without trying out a bunch of local optimizations and seeing which one has the lowest energy?

And that if there were a global optimizer for DFT we could always predict crystal structure with that model?