  We study the role of renormalon cancellation schemes and perturbative scale
choices in extractions of the strong coupling constant $\alpha_s(m_Z)$ and the
leading non-perturbative shift parameter $\Omega_1$ from resummed predictions
of the $e^+e^-$ event shape thrust. We calculate the thrust distribution to
N$^{3}$LL$^\prime$ resummed accuracy in Soft-Collinear Effective Theory (SCET)
matched to the fixed-order $\mathcal{O}(\alpha_s^2)$ prediction, and perform a
new high-statistics computation of the $\mathcal{O}(\alpha_s^3)$ matching in
EERAD3, although we do not include the latter in our final $\alpha_s$ fits due
to some observed systematics that require further investigation. We are
primarily interested in testing the phenomenological impact sourced from
varying amongst three renormalon cancellation schemes and two sets of
perturbative scale profile choices. We then perform a global fit to available
data spanning center-of-mass energies between 35-207 GeV in each scenario.
Relevant subsets of our results are consistent with prior SCET-based
extractions of $\alpha_s(m_Z)$, but we are also led to a number of novel
observations. Notably, we find that the combined effect of altering the
renormalon cancellation scheme and profile parameters can lead to
few-percent-level impacts on the extracted values in the $\alpha_s-\Omega_1$
plane, indicating a potentially important systematic theory uncertainty that
should be accounted for. We also observe that fits performed over windows
dominated by dijet events are typically of a higher quality than those that
extend into the far tails of the distributions, possibly motivating future fits
focused more heavily in this region. Finally, we discuss how different
estimates of the three-loop soft matching coefficient $c_{\tilde{S}}^3$ can
also lead to measurable changes in the fitted $\lbrace \alpha_s, \Omega_1
\rbrace$ values.
