We probe the mechanisms at work in the build-up of thermoreversible gel networks, with the help of hybrid gelatin gels containing a controlled density of irreversible, covalent crosslinks (CLs), which we quench below the physical gelation temperature. The detailed analysis of the dependence on covalent crosslink density of both the shear modulus and optical activity evolutions with time after quench enables us to identify two stages of the physical gelation process, separated by a temperature-dependent crossover modulus: (i) an early nucleation regime during which rearrangements of the triple-helix CLs play a negligible role, and (ii) a late, logarithmic aging one, which is preserved, though slowed down, in the presence of irreversible CLs. We show that aging is fully controlled by rearrangements and discuss the implication of our results in terms of the switch from an early, local dynamics to a late, cooperative long-range one.