  The $230$ GHz lightcurves of Sagittarius~A* (Sgr~A*) predicted by general
relativistic magnetohydrodynamics (GRMHD) and ray-tracing (GRRT) models in
Event Horizon Telescope Collaboration et al. (2022) have higher variability
$M_{\Delta T}$ compared to observations. In this series of papers, we explore
the origin of such large brightness variability. In this first paper, we
performed large GRRT parameter surveys that span from the optically thin to the
optically thick regimes, covering the ion-electron temperature ratio under
strongly magnetized conditions, $R_{\rm Low}$, from $1$ to $60$. We find that
increasing $R_{\rm Low}$ can lead to either an increase or a reduction in
$M_{\Delta T}$ depending on other model parameters, making it consistent with
the observed variability of Sgr~A* in some cases. Our analysis of GRRT image
snapshots finds that the major contribution to the large $M_{\Delta T}$ for the
$R_{\rm Low} = 1$ models comes from the photon rings. However, secondary
contributions from the accretion flow are also visible depending on the spin
parameter. Our work demonstrates the importance of the electron temperature
used for modelling radiatively inefficient accretion flows and places new
constraints on the ion-electron temperature ratio. A more in-depth analysis for
understanding the dependencies of $M_{\Delta T}$ on $R_{\rm Low}$ will be
performed in subsequent papers.
