The well-established correlations between the mass of a galaxy and the properties of its stars are considered evidence for mass driving the evolution of the stellar population. However, for early-type galaxies (ETGs), we find that $g-i$ color and stellar metallicity [Z/H] correlate more strongly with gravitational potential $Phi$ than with mass $M$, whereas stellar population age correlates best with surface density $Sigma$. Specifically, for our sample of 625 ETGs with integral-field spectroscopy from the SAMI Galaxy Survey, compared to correlations with mass, the color--$Phi$, [Z/H]--$Phi$, and age--$Sigma$ relations show both smaller scatter and less residual trend with galaxy size. For the star formation duration proxy [$alpha$/Fe], we find comparable results for trends with $Phi$ and $Sigma$, with both being significantly stronger than the [$alpha$/Fe]-$M$ relation. In determining the strength of a trend, we analyze both the overall scatter, and the observational uncertainty on the parameters, in order to compare the intrinsic scatter in each correlation. These results lead us to the following inferences and interpretations: (1) the color--$Phi$ diagram is a more precise tool for determining the developmental stage of the stellar population than the conventional color--mass diagram; and (2) gravitational potential is the primary regulator of global stellar metallicity, via its relation to the gas escape velocity. Furthermore, we propose the following two mechanisms for the age and [$alpha$/Fe] relations with $Sigma$: (a) the age--$Sigma$ and [$alpha$/Fe]--$Sigma$ correlations arise as results of compactness driven quenching mechanisms; and/or (b) as fossil records of the $Sigma_{SFR}proptoSigma_{gas}$ relation in their disk-dominated progenitors.
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