Uniform Forward-Modeling Analysis of Ultracool Dwarfs. IV. Benchmarking the Sonora Diamondback and Saumon & Marley (2008) Atmospheric Models Across Late-M, L, and T types with Low-Resolution 0.8-2.5 $\mu$m Spectroscopy
(Abridged) We present a systematic assessment of two major cloudy atmospheric model grids -- SM08 (Saumon & Marley 2008) and Sonora Diamondback -- when applied to low-resolution near-infrared (0.8-2.5 $\mu$m) spectroscopy. Our analysis focuses on a uniform sample of 142 age-benchmark brown dwarfs and planetary-mass objects spanning late-M, L, and T spectral types, with independently determined ages from 10 Myr to 10 Gyr. We perform forward-model spectral fitting for all benchmarks' IRTF/SpeX spectra ($R\sim$80-250) using both SM08 and Sonora Diamondback atmospheric models to infer effective temperatures, surface gravities, metallicities, radii, and cloud sedimentation efficiencies. The two model grids yield broadly consistent results. Among L4-L9 dwarfs, we identify a statistically significant, population-level age dependence of the cloud parameter $f_{\rm sed}$, with young benchmarks ($<300$ Myr) exhibiting systematically lower $f_{\rm sed}$ values than older counterparts. This trend is absent across L0-T5 and T0-T5, demonstrating that cloud properties vary with age and surface gravity and offering explanations for the observed gravity-dependent photometric properties at the late-L end of the L/T transition. By comparing spectroscopically inferred parameters with predictions from evolution models, we quantify systematic errors in the fitted atmospheric parameters and establish empirical calibrations to anchor future studies using these atmospheric models. Stacked residuals of the sample reveal wavelength-dependent data-model mismatches associated with key atomic and molecular absorption bands, highlighting the need for improved opacities and rainout chemistry. Finally, we show that including an interstellar-medium-like extinction term significantly improves the spectral fits, confirming and broadening previous findings and suggesting missing opacity sources in current cloudy models.