But the integrals are derived from a theoretical set of equations. You could develop a strictly numerical approach based on the original equations -- thus integrals would never be utilized. The integrals are typically derived from a perturbation expansion of the original *physical* equations.
Feynman diagrams are used in QED, quantum electrodynamics, they represent all possible interactions that may occur. A path integral is used to find the minimum energy for each interaction. The sum of these interaction energies is the total E. Without writing down the possible interactions of any particular situation the original theoretical eq. is useless, because the solution involves finding the energy that is a sum of all possible interactions. Without knowing the terms in the sum the actual sum can never be totaled. Each term in the sum involves finding the minimum energy for the particular interaction, that's where the path integral, or it's equivalent enters as a necessity.