How does a galaxy's quenching probability depend on its black hole's mass?

We develop a parametric toy model to explore how black hole mass scatter ${\sigma }_{\mathrm{B}\mathrm{H}}$ and the dependence of the quenching probability per unit time on black hole mass, parameterized as a power law with index $q$, jointly control the faint-end slope $\alpha$ of the stellar mass function of quenched central galaxies. By varying ${\sigma }_{\mathrm{B}\mathrm{H}}$ and $q$, we identify two regimes. At low scatter, ${\sigma }_{\mathrm{B}\mathrm{H}}\lesssim 0.4$ dex, $\alpha$ is strongly sensitive to $q$. Gradual quenching, represented by low $q$, produces shallow slopes, while sharp thresholds, with high $q$, produce steep mass functions, making $\alpha$ a powerful diagnostic of AGN feedback prescriptions. At large scatter, ${\sigma }_{\mathrm{B}\mathrm{H}}\gtrsim 0.6$ dex, models with vastly different $q$ converge to similar quenched mass functions, because the intrinsic scatter erases the imprint of the feedback implementation. Comparing to cosmological hydrodynamical simulations and semianalytic models, only models with ${\sigma }_{\mathrm{B}\mathrm{H}}\gtrsim 0.4$ dex match the observed faint-end slope of $\alpha \approx 0.7$. {\sc SAGE}, {\sc Dark Sage}, and {\sc COLIBRE}, with ${\sigma }_{\mathrm{B}\mathrm{H}}$ of 2.0, 0.6, and 0.4 dex, respectively, reproduce this slope, whereas {\sc IllustrisTNG} and {\sc EAGLE}, with ${\sigma }_{\mathrm{B}\mathrm{H}}\lesssim 0.2$ dex, produce steeper slopes. Current observational estimates of ${\sigma }_{\mathrm{B}\mathrm{H}}$ allow a wide range of $q\gtrsim 0.8$, so the observed slope alone cannot identify the quenching prescription. If ${\sigma }_{\mathrm{B}\mathrm{H}}$ is small, simultaneous measurements of $\alpha$ and ${\sigma }_{\mathrm{B}\mathrm{H}}$ can distinguish between competing AGN feedback models.