When a charged surface, such as an electrode, colloid, or protein, is submerged into an electrolyte or ionic liquid, ions within the fluid rearrange into electric double layers (EDLs) that electrostatically screen the interfacial charge. The electrostatic potential and ion distributions within EDLs have long been described by mean-field local-density approximations (LDAs) that assume flat electrodes, uncorrelated ions, and bulk forms for the chemical potential. The objective of this work is to elucidate LDA failure mechanisms and supplement or supplant mean-field treatments of electrochemical systems that fail to capture correlated behavior. We develop an exceedingly general method, which requires no a priori model and identifies whether EDLs in a given electrolyte can obey a LDA, or whether more advanced approaches (e.g. integro-differential equations, atomistic simulations, etc.) are required, irrespective of the source of LDA breakdown. We combine continuum-level theoretical studies with complementary simulations in order to critically assess the accuracy of LDA models of implicit solvent electrolytes with equal and differently sized ions. We also pose a novel LDA model that seeks to address solvation, polarizability, and finite-size interactions present in actual and simulated EDLs with explicit solvent.