The broadband and ultrafast photoresponse of graphene has been extensively studied in recent years, although the photoexcited carrier dynamics is still far from being completely understood. Different experimental approaches imply either one of two fundamentally different scattering mechanisms for hot electrons. One is high-energy optical phonons, while the other is disorder-driven supercollisions with acoustic phonons. However, the concurrent relaxation via both optical and acoustic phonons has not been considered so far, hindering the interpretation of different experiments within a unified framework. Here we expand the optical phonon-mediated cooling model, to include electron scattering with the acoustic phonons. By assuming the enhancement of electron-acoustic phonon supercollisions from the localized defect at the photothermoelectric current-generating interface, we provide a broader perspective to the ultrafast photoresponse of graphene, highlighting the previously overlooked effect of the interface for cooling dynamics. We show that the transient photothermoelectric response, which has been attributed exclusively to supercollisions, can be successfully explained without rejecting the established optical phonon relaxation pathway, demonstrating that the two cooling mechanisms are not mutually exclusive but complement each other.