  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 datasets from 21-cm, cosmic microwave background, and galaxy survey
measurements. Though a relatively simple picture of high-$z$ galaxy evolution
has emerged in recent years that should be well constrained by upcoming
observations, there are many potential sources of systematic uncertainty in
models that could bias and/or degrade these 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, the halo mass function (HMF), and the choice of stellar
population synthesis (SPS) model. We find that cosmological uncertainties are
small, affecting the Thomson scattering optical depth at the few percent level
and the amplitude of the global 21-cm signal at the $\sim$few mK level. The
differences brought about by choice of HMF and SPS models are more dramatic,
comparable to the $1 \sigma$ error-bar on $\tau_e$ and a $\sim 20$ mK effect on
the global 21-cm signal amplitude. We compare models with comparable empirical
predictions by jointly fitting galaxy luminosity functions and global 21-cm
signals, 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 the uncertainties in common modeling ingredients in
order to enable precision inference with future datasets.
