We formulate a new closure approximation (triple-modified Verlet (MV)) for the polymer integral equation theory (PRISM) of microstructure and miscibility of dense polymer nanocomposites (PNC). Complementary new molecular dynamics simulations of the same model PNC are also performed over a wide range of system parameters, including total mixture packing fraction, nanoparticle (NP) loading, and NP-polymer interfacial attraction strength. The three distinct states of NP organization (entropic depletion clustering, adsorbed layers and steric stabilization, and bridging polymer-NP network formation) previously predicted are observed in our simulations. For the location and shape of phase boundaries, PRISM theory with the new triple-MV closure qualitatively and sometimes quantitatively agrees with our simulations, which is consistent with conclusions we recently drew for dense colloidal sphere mixtures with large size asymmetry. PRISM-MV predictions for both the spinodal demixing boundaries and the three distinct real space intermolecular site-site pair correlations are generally found to be much more accurate than those based on the classic mixed closure that employs Percus-Yevick (PY) and hypernetted chain (HNC) closure approximations, especially at low to intermediate NP loadings of central interest in polymer science. For the local structure we find both closures perform well at very high NP loading. However, at low and intermediate NP loadings, PRISM theory results based on the triple-MV closure are in much better agreement with simulations, including the rigorous positivity of all pair correlation functions even under strong interfacial attraction conditions. Beyond using our simulations to test theory, we discuss new polymer physics insights gleaned from examination of the computational results in detail. The improved structural theory also provides foundational input for developing predictive dynamical theories.