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The Galactic Center GeV excess (GCE) has garnered great interest as a possible signal of either dark matter annihilation or some novel astrophysical phenomenon, such as a new population of gamma-ray emitting pulsars. In a companion paper, we showed that in a $10^\circ$ radius region of interest (ROI) surrounding the Galactic Center, apparent evidence for GCE point sources (PSs) from non-Poissonian template fitting (NPTF) is actually an artifact of unmodeled north-south asymmetry of the GCE. In this work, we develop a simplified analytic description of how signal mismodeling can drive an apparent preference for a PS population, and demonstrate how the behavior pointed out in the companion paper also appears in simpler simulated datasets that contain no PS signals at all. We explore the generality of this behavior in the real gamma-ray data, and discuss the implications for past and future studies using NPTF techniques. While the drop in PS preference once north-south asymmetry is included is not ubiquitous in larger ROIs, we show that any overly-rigid signal model is expected to yield a spurious PS signal that can appear very convincing: as well as apparent significance comparable to what one would expect from a true PS population, the signal can exhibit stability against a range of variations in the analysis, and a source count function that is very consistent with previous apparent NPTF-based detections of a GCE PS population. This contrasts with previously-studied forms of systematic mismodeling which are unlikely to mimic a PS population in the same way. In the light of this observation, and its explicit realization in the region where the GCE is brightest, we argue that a dominantly smooth origin for the GCE is not in tension with existing NPTF analyses.