  Type Ia supernovae (SNe Ia) are standardizable candles that must be modeled
empirically to yield cosmological constraints. To understand the robustness of
this modeling to variations in the model training procedure, we build an
end-to-end pipeline to test the recently developed SALT3 model. We explore the
consequences of removing pre-2000s low-$z$ or poorly calibrated $U$-band data,
adjusting the amount and fidelity of SN Ia spectra, and using a
model-independent framework to simulate the training data. We find the SALT3
model surfaces are improved by having additional spectra and $U$-band data, and
can be shifted by $\sim 5\%$ if host galaxy contamination is not sufficiently
removed from SN spectra. We find that resulting measurements of $w$ are
consistent to within $2.5\%$ for all training variants explored in this work,
with the largest shifts coming from variants that add color-dependent
calibration offsets or host galaxy contamination to the training spectra, and
those that remove pre-2000s low-$z$ data. These results demonstrate that the
SALT3 model training procedure is largely robust to reasonable variations in
the training data, but that additional attention must be paid to the treatment
of spectroscopic data in the training process. We also find that the training
procedure is sensitive to the color distributions of the input data; the
resulting $w$ measurement can be biased by $\sim2\%$ if the color distribution
is not sufficiently wide. Future low-$z$ data, particularly $u$-band
observations and high signal-to-noise ratio SN Ia spectra, will help to
significantly improve SN Ia modeling in the coming years.
