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Models of the reionization and reheating of the intergalactic medium (IGM) at redshifts $z \gtrsim 6$ continue to grow more sophisticated in anticipation of near-future 21-cm, cosmic microwave background, and galaxy survey measurements. However, there are many potential sources of systematic uncertainty in models that could bias and/or degrade upcoming constraints if left unaccounted for. In this work, we examine three commonly-ignored sources of uncertainty in models for the mean reionization and thermal histories of the IGM: the underlying cosmology, halo mass function (HMF), and choice of stellar population synthesis (SPS) model. We find that cosmological uncertainties affect the Thomson scattering optical depth at the few percent level and the amplitude of the global 21-cm signal at the $\sim$5-10 mK level. The differences brought about by choice of HMF and SPS models are more dramatic, comparable to the $1 σ$ error-bar on $τ_e$ and a $\sim 20$ mK effect on the global 21-cm signal amplitude. Finally, we jointly fit galaxy luminosity functions and global 21-cm signals for all HMF/SPS combinations and find that (i) doing so requires additional free parameters to compensate for modeling systematics and (ii) the spread in constraints on parameters of interest for different HMF and SPS choices, assuming $5$ mK noise in the global signal, is comparable to those obtained when adopting the "true" HMF and SPS with $\gtrsim 20$ mK errors. Our work highlights the need for dedicated efforts to reduce modeling uncertainties in order to enable precision inference with future datasets.