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The Fermi Large Area Telescope has observed an excess of ~GeV energy gamma rays from the center of the Milky Way, which may arise from near-thermal dark matter annihilation. Firmly establishing the dark matter origin for this excess is however complicated by challenges in modeling diffuse cosmic-ray foregrounds as well as unresolved astrophysical sources, such as millisecond pulsars. Non-Poissonian Template Fitting (NPTF) is one statistical technique that has previously been used to show that at least some fraction of the GeV excess is likely due to a population of dim point sources. These results were recently called into question by Leane and Slatyer (2019), who showed that a synthetic dark matter annihilation signal injected on top of the real Fermi data is not recovered by the NPTF procedure. In this work, we perform a dedicated study of the Fermi data and explicitly show that the central result of Leane and Slatyer (2019) is likely driven by the fact that their choice of model for the Galactic foreground emission does not provide a sufficiently good description of the data. We repeat the NPTF analyses using a state-of-the-art model for diffuse gamma-ray emission in the Milky Way and introduce a novel statistical procedure, based on spherical-harmonic marginalization, to provide an improved description of the Galactic diffuse emission in a data-driven fashion. With these improvements, we find that the NPTF results continue to robustly favor the interpretation that the Galactic Center excess is due, in part, to unresolved astrophysical point sources across the analysis variations that we have explored.