We present an alternative approach to the theory of free Gibbs laws with convex potentials developed by Dabrowski, Guionnet, and Shlyakhtenko. Instead of solving SDE's, we combine PDE techniques with a notion of asymptotic approximability by trace polynomials for a sequence of functions on $M_N(\mathbb{C})_{sa}^m$ to prove the following. Suppose $\mu_N$ is a probability measure on on $M_N(\mathbb{C})_{sa}^m$ given by uniformly convex and semi-concave potentials $V_N$, and suppose that the sequence $DV_N$ is asymptotically approximable by trace polynomials in a certain sense. Then the moments of $\mu_N$ converge to a non-commutative law $\lambda$. Moreover, the free entropies $\chi(\lambda)$, $\underline{\chi}(\lambda)$, and $\chi^*(\lambda)$ agree and equal the limit of the normalized classical entropies of $\mu_N$. An upcoming paper will use the same techniques to obtain transport maps from $\lambda$ to a free semicircular family as the limit of transport maps for the matrix models $\mu_N$.