Why density functional theory favors a uniform electron density? Why density functional theory (DFT) favors a uniform electron density when we use it to calculate ground state electronic structure self-consistently? 

For example, when we use DFT for calculating the (ground state) electron density of two $\text{H}$ atoms with a separation somewhat larger then the bond length of an $\text{H}_2$ molecule, there will still be a non-negligible electron density between the two atoms as if there is still a covalent bond. As a result, the calculated electronic ground state is essentially an excited state of the system. 

I cannot remember where did I read this, and I don't understand why DFT has such a problem. I think the problem resides in the fact that we use a local exchange-correlation functional $E_{XC}$ like LDA and GGA. But I don't understand why a local $E_{XC}$ will lead to a uniform electron density.
I think I can see the confusion, and I will do my best to explain.

As an initial attempt to use the electron density, the uniform electron gas (UEG) model was developed. This fictitious model assumes: an infinite number of electrons, a constant non-zero density, an infinite volume, and a uniformly distributed positive charge. IMPORTANT: The electron-electron interactions (correlation and exchange) are fundamentally ignored.

Two main theorems comprise modern DFT: Hohenberg-Kohn (HK) and Kohn-Sham (KS).
In principle DFT is exact if we knew the universal HK functional (which contains electron-electron exchange-correlation terms); however, no mathematical description has been developed. As we do not know the exact form of the exchange-correlation term, a number of exchange-correlation functional have been developed. These can be derived from experimental thermochemical date or high-accuracy quantum calculations. 

As exchange-correlation functionals are approximations, the electron density is not fully localised to a specific area. I hope this helps.