Using a geometry-based fundamental measure density functional theory, we calculate bulk fluid phase diagrams of colloidal mixtures of vanishingly thin hard circular platelets and hard spheres. We find isotropic-nematic phase separation, with strong broadening of the biphasic region, upon increasing the pressure. In mixtures with large size ratio of platelet and sphere diameters, there is also demixing between two nematic phases with differing platelet concentrations. We formulate a fundamental measure density functional for mixtures of colloidal platelets and freely overlapping spheres, which represent ideal polymers, and use it to obtain phase diagrams. We find that, for low platelet-polymer size ratio, in addition to isotropic-nematic and nematic-nematic phase coexistence, platelet-polymer mixtures also display isotropic-isotropic demixing. By contrast, we do not find isotropic-isotropic demixing in hard-core platelet-sphere mixtures for the size ratios considered.