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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.