Interfacial Rayleigh-Taylor mixing is crucial to describing important natural and engineering processes such as exploding supernovae, laser micro-machining, hot spots in inertial confinement fusion, and optical telecommunications. These require the characterization of the time dependence of the driving acceleration. We compare our theoretical formulation based on group theory foundations with interface-capturing numerical simulations for linear and nonlinear two-dimensional Rayleigh-Taylor instabilities in a finite-sized domain with time-varying acceleration over broad ranges of Atwood numbers and acceleration exponents. Detailed corroboration between theory and simulations is provided for this foundational case. Both demonstrate the strong interfacial nature of Rayleigh-Taylor instabilities, which suggests that practical flow fields can be reconstructed from the derived fluid potential using the proposed theory. Robust agreement is also obtained for the early- and late-time evolution of the amplitudes of the bubble and spike, which demonstrate that the Rayleigh-Taylor flow can transition to the mixing regime even for a single-mode initial perturbation. Corroboration with experiments in high energy density plasmas motivated by studies of supernovae is also achieved. In addition, a long-standing puzzle in Rayleigh-Taylor dynamics on the interplay between shear and curvature in theory and simulations is resolved by accounting for finite viscosity in the latter. The characterization of Rayleigh-Taylor instabilities as a highly interfacial phenomenon provides valuable insights into its multi-scale nature, which enhances the design and understanding of numerous processes of practical interest.