Born as ice-rich planetesimals, cometary nuclei were gravitationally scattered onto their current orbits in the Kuiper Belt and the Oort Cloud during the giant planets' dynamical instability in the early stages of our Solar System's history. Here, we model the thermal evolution of planetesimals during and after the giant planet instability. We couple an adapted thermal evolution model to orbital trajectories provided by \textit{N}-body simulations to account for the planetesimals' orbital evolution, a parameter so far neglected by previous thermal evolution studies. Our simulations demonstrate intense thermal processing in all planetesimal populations, concerning mainly the hyper-volatile ice content. Unlike previous predictions, we show that hyper-volatile survival was possible in a significant number of planetesimals of the Kuiper Belt and the Oort Cloud. Planetesimals ejected into the interstellar space proved to be the most processed, while planetesimals ending in the Oort Cloud were the least processed population. We show that processing differences between populations are a direct consequence of their orbital evolution patterns, and that they provide a natural explanation for the observed variability in the abundance ratios of CO on cometary populations and on the recent observations of long-distance CO-driven activity on inbound Long-period Comets.