The potential of lead-halide perovskites for realistic applications is currently hindered by their limited long-term stability under functional activation. While the role of lattice flexibility in the thermal response of perovskites has become increasingly evident, the description of thermally-induced distortions is still unclear. In this work, we provide a unified picture of thermal activation in CsPbBr₃ across length scales, showing that lattice symmetry does not increase at high temperatures. We combine temperature-dependent XRD, Br K-edge XANES, ab initio MD simulations, and calculations of the XANES spectra by first-principles, accounting for both thermal fluctuations and core hole final state effects. We find that the octahedral tilting of the Pb-Br inorganic framework statistically adopts multiple local configurations over time - in the short-range. In turn, the stochastic nature of the local thermal fluctuations uplifts the longer-range periodic octahedral tilting characterizing the low temperature structure, with the statistical mean of the local configurations resulting in a cubic-like time-averaged lattice. These observations can be rationalized in terms of displacive thermal phase transitions through the soft mode model, in which the phonon anharmonicity of the flexible inorganic framework causes the excess free energy surface to change as a function of temperature. Our work demonstrates that the effect of thermal dynamics on the XANES spectra can be effectively described for largely anharmonic systems, provided ab initio MD simulations are performed to determine the dynamically fluctuating structures, and core hole final state effects are included in order to retrieve an accurate XANES line shape. Moreover, it shows that the soft mode model, previously invoked to describe displacive thermal phase transitions in oxide perovskites, carries a more general validity.