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 σ BH 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 α of the stellar mass function of quenched central galaxies. By varying σ BH and q, we identify two regimes. At low scatter, σ BH 0.4 dex, α 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 α a powerful diagnostic of AGN feedback prescriptions. At large scatter, σ BH 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 σ BH 0.4 dex match the observed faint-end slope of α about 0.7. SAGE, Dark Sage, and COLIBRE, with σ BH of 2.0, 0.6, and 0.4 dex, respectively, reproduce this slope, whereas IllustrisTNG and EAGLE, with σ BH 0.2 dex, produce steeper slopes. Current observational estimates of σ BH allow a wide range of q 0.8, so the observed slope alone cannot identify the quenching prescription. If σ BH is small, simultaneous measurements of α and σ BH can distinguish between competing AGN feedback models.