Three-dimensional (3D) simulations in recent years have shown severe difficulties producing 10^51 erg explosions of massive stars with neutrino based mechanisms while on the other hand demonstrated the large potential of mechanical effects, such as winds and jets in driving explosions. In this paper we study the typical time-scale and energy for accelerating gas by neutrinos in core-collapse supernovae (CCSNe) and find that under the most extremely favorable (and probably unrealistic) conditions, the energy of the ejected mass can reach at most 5X10^50 erg. More typical conditions yield explosion energies an order-of-magnitude below the observed 10^51 erg explosions. On the other hand, non-spherical effects with directional outflows hold promise to reach the desired explosion energy and beyond. Such directional outflows, which in some simulations are produced by numerical effects of 2D grids, can be attained by angular momentum and jet launching. Our results therefore call for a paradigm shift from neutrino-based explosions to jet-driven explosions for CCSNe.