The local-operator entanglement (LOE) measures the classical simulability of a Heisenberg operator and is conjectured to witness many-body chaos in locally interacting systems. Using tools from free probability, we analytically compute its asymptotic value for Haar random dynamics for all Rényi indices. We find that to the first two orders in large dimension, for traceless operators it exactly reproduces the Page curve for random states. In contrast to higher-order out-of-time-ordered correlators (OTOCs), which depend on more complex operator correlations via free cumulants, the leading-order LOE is independent of the initial operator. We detail exactly how these higher-order correlations are encoded in the coefficients of the finite-size corrections. Guided by our Haar result, we argue that the long-time value of the LOE entropies in chaotic systems is robust: they depend only on a simple sub-sector of the hierarchy of correlations encoded by the full Eigenstate Thermalization Hypothesis. Our results therefore provide a conceptual boundary between operator simulability and higher-order scrambling diagnostics.