Most simulations of outflow feedback on star formation are based on the assumption that outflows are driven by a wide angle "Xwind," 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 yrs. 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 α 90 • . Third, at realistic ages of ∼10 000 yrs, 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.