The Kekulé order in graphene is known to give rise to extended flat bands near the van Hove singularity (VHS) and enhanced electron-phonon coupling. However, the microscopic mechanism underlying the coexistence of these effects remains unclear. Here, it is shown that they jointly emerge as a consequence of the lattice distortion manifested through the differentiation of carbon bonds and associated electronic hoppings. In this picture, the conduction band around the M point decreases almost linearly in width with increasing Kekulé order, while the VHS and dimensionless electron-phonon coupling exhibit approximately quadratic upward responses. The resulting enhancement of the electron-phonon coupling is governed predominantly by the increasing density of states at the VHS, with the effective deformation potential remaining weakly affected by the distortion. As such, the obtained results are consistent with previous related theoretical and experimental studies, while also pointing toward a potential shallowing of the conduction band under the characteristic 1/4 filling and an increasing relevance of nonadiabatic electron-phonon effects beyond the conventional Migdal approximation. Altogether, these findings highlight the potential of controlled Kekulé ordering for engineering emergent correlated quantum phases in graphene.