Magnetization manipulation is essential for basic research and applications. A fundamental question is, how fast can the magnetization be reversed in nanoscale magnetic storage media. When subject to an ultrafast laser pulse, the speed of the magnetization dynamics depends on the nature of the energy transfer pathway. The order of the spin system can be effectively influenced through spin-flip processes mediated by hot electrons. It has been predicted that as electrons drive spins into the regime close to almost total demagnetization, characterized by a loss of ferromagnetic correlations near criticality, a second slower demagnetization process takes place after the initial fast drop of magnetization. By studying FePt, we unravel the fundamental role of the electronic structure. As the ferromagnet Fe becomes more noble in the FePt compound, the electronic structure is changed and the density of states around the Fermi level is reduced, thereby driving the spin correlations into the limit of critical fluctuations. We demonstrate the impact of the electrons and the ferromagnetic interactions, which allows a general insight into the mechanisms of spin dynamics when the ferromagnetic state is highly excited, and identifies possible recording speed limits in heat-assisted magnetization reversal.