  Asteroseismic modelling is a powerful way to derive stellar properties.
However, the derived quantities are limited by built-in assumptions used in
stellar models. This work presents a detailed characterisation of stellar model
uncertainties in asteroseismic red giants, focusing on the mixing-length
parameter $\alpha_{\rm MLT}$, the initial helium fraction $Y_{\rm init}$, the
solar abundance scale, and the overshoot parameters. First, we estimate error
floors due to model uncertainties to be $\approx$0.4\% in mass, $\approx$0.2\%
in radius, and $\approx$17\% in age, primarily due to the uncertain state of
$\alpha_{\rm MLT}$ and $Y_{\rm init}$. The systematic uncertainties in age
exceed typical statistical uncertainties, suggesting the importance of their
evaluation in asteroseismic applications. Second, we demonstrate that the
uncertainties from $\alpha_{\rm MLT}$ can be entirely mitigated by direct
radius measurements or partially through $\nu_{\rm max}$. Utilizing radii from
Kepler eclipsing binaries, we determined the $\alpha_{\rm MLT}$ values and
calibrated the $\alpha_{\rm MLT}$--[M/H] relation. The correlation observed
between the two variables is positive, consistent with previous studies using
1-D stellar models, but in contrast with outcomes from 3-D simulations. Third,
we explore the implications of using asteroseismic modelling to test the
$\nu_{\rm max}$ scaling relation. We found that a perceived dependency of
$\nu_{\rm max}$ on [M/H] from individual frequency modelling can be largely
removed by incorporating the calibrated $\alpha_{\rm MLT}$--[M/H] relation.
Variations in $Y_{\rm init}$ can also affect $\nu_{\rm max}$ predictions. These
findings suggest that $\nu_{\rm max}$ conveys information not fully captured by
individual frequencies, and that it should be carefully considered as an
important observable for asteroseismic modelling.
