  The relation between the integrated thermal Sunyaev-Zeldovich (tSZ)
$y$-decrement versus halo mass ($Y$--$M$) can potentially constrain galaxy
formation models, if theoretical and observational systematics can be properly
assessed. We investigate the $Y$--$M$ relation in the SIMBA and
IllustrisTNG-100 cosmological hydrodynamic simulations, quantifying the effects
of feedback, line-of-sight projection, and beam convolution. We find that
SIMBA's AGN jet feedback generates strong deviations from self-similar
expectations for the $Y$--$M$ relation, especially at
$M_{500}<10^{13}M_{\odot}$. In SIMBA, this is driven by suppressed in-halo $y$
contributions owing to lowered halo baryon fractions. IllustrisTNG results more
closely resemble SIMBA without jets. Projections of line-of-sight structures
weaken these model differences slightly, but they remain significant --
particularly at group and lower halo masses. In contrast, beam smearing at
$\textit{Planck}$ resolution makes the models indistinguishable, and both
models appear to agree well with $\textit{Planck}$ data down to the lowest
masses probed. We show that the arcminute resolution expected from forthcoming
facilities would retain the differences between model predictions, and thereby
provide strong constraints on AGN feedback.
