Most simulations of outflow feedback on star formation are based on the assumption that outflows are driven by a wide angle “X-wind,” rather than a narrow jet. However, the arguments initially raised against pure jet-driven flows were based on steady ejection in a uniform medium, a notion that is no longer supported based on recent observations. We aim to determine whether a pulsed narrow jet launched in a density-stratified, self-gravitating core could reproduce typical molecular outflow properties, without the help of a wide-angle wind component. We performed axisymmetric hydrodynamic simulations using the MPI-AMRVAC code with optically thin radiative cooling and grid refinement down to 5 au, on timescales up to 10 000 yr. Then we computed the predicted properties for the purposes of a comparison with observational data. First, the jet-driven shell expands much faster and wider through a core with steeply decreasing density than through an uniform core. Second, when blown into the same singular flattened core, a jet-driven shell shows a similar width as a wide-angle wind-driven shell in the first few hundred years, but a decelerating expansion on long timescales. The flow adopts a conical shape, with a sheared velocity field along the shell walls and a base opening angle reaching up to a ≃ 90°. Third, at realistic ages of ~10 000 yr, a pulsed jet-driven shell shows fitting features along with a qualitative resemblance with recent observations of protostellar outflows with the Atacama Large Millimeter Array, such as HH46–47 and CARMA–7. In particular, similarities can be seen in the shell widths, opening angles, position-velocity diagrams, and mass-velocity distribution, with some showing a closer resemblance than in simulations based on a wide-angle “X-wind” model. Therefore, taking into account a realistic ambient density stratification in addition to millenia-long integration times is equally essential to reliably predict the properties of outflows driven by a pulsed jet and to confront them with the observations.Key words: stars: formation / stars: pre-main sequence / methods: numerical / ISM: jets and outflows / shock waves / hydrodynamics