We study the $\nabla φ$ model with uniformly convex Hamiltonian $\mathcal{H} (φ) := \sum V(\nabla φ)$ and prove a quantitative rate of convergence for the finite-volume surface tension as well as a quantitative rate estimate for the $L^2$-norm for the field subject to affine boundary condition. One of our motivations is to develop a new toolbox for studying this problem that does not rely on the Helffer-Sjöstrand representation. Instead, we make use of the variational formulation of the partition function, the notion of displacement convexity from the theory of optimal transport, and the recently developed theory of quantitative stochastic homogenization.