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Thermal emission has now been observed from many dozens of exoplanet atmospheres, opening the gateway to population-level characterization. Here, we provide theoretical explanations for observed trends in $\textit{Spitzer}$ IRAC channel 1 (3.6 $μm$) and channel 2 (4.5 $μm$) photometric eclipse depths (EDs) across a population of 34 hot Jupiters. We apply planet-specific, self-consistent atmospheric models, spanning a range of recirculation factors, metallicities, and C/O ratios, to probe the information content of $\textit{Spitzer}$ secondary eclipse observations across the hot-Jupiter population. We show that most hot Jupiters are inconsistent with blackbodies from $\textit{Spitzer}$ observations alone. We demonstrate that the majority of hot Jupiters are consistent with low energy redistribution between the dayside and nightside (hotter dayside than expected with efficient recirculation). We also see that high equilibrium temperature planets (T$_{eq}$ $\ge$ 1800 K) favor inefficient recirculation in comparison to the low temperature planets. Our planet-specific models do not reveal any definitive population trends in metallicity and C/O ratio with current data precision, but more than 59 % of our sample size is consistent with the C/O ratio $\leq$ 1 and 35 % are consistent with whole range (0.35 $\leq$ C/O $\leq$ 1.5). We also find that for most of the planets in our sample, 3.6 and 4.5 $μm$ model EDs lie within $\pm$1 $σ$ of the observed EDs. Intriguingly, few hot Jupiters exhibit greater thermal emission than predicted by the hottest atmospheric models (lowest recirculation) in our grid. Future spectroscopic observations of thermal emission from hot Jupiters with the James Webb Space Telescope will be necessary to robustly identify population trends in chemical compositions with its increased spectral resolution, range and data precision.