Pulsars are highly magnetized rotating neutron stars, emitting in a broad electromagnetic energy range. Reproducing the observed population of pulsars refines our understanding of their formation, evolution as well as radiation processes and geometry. In this paper, we improve our previous pulsar population synthesis by studying the impact of the Galactic gravitational potential and of the death line putting emphasis on the gamma-ray pulsar population. In order to elucidate the necessity of a death line, refined initial distributions for spin periods and radial positions at birth were implemented, elevating the sophistication of our previous simulations to the most recent state-of-the-art. Our pulsar population synthesis takes into account the secular evolution of a force-free magnetosphere and the magnetic field decay. The radio and gamma-ray emissions are modeled respectively by the polar cap geometry and the striped wind model. When simulating a cohort of ten million pulsars akin to previous endeavors, the integration of a death line proves essential for a better agreement with observational trends. Yet, discrepancies emerges when modeling a reduced ensemble of only one million pulsars, endowed with a higher birth rate. This divergence yields a remarkably refined $P-\dot{P}$ diagram, conspicuously with or without any delineating death line. Intriguingly, this observation suggests an overestimation of the ages of detected pulsars in existent data sets, thereby challenging the necessity of death lines in pulsar population synthesis. Validation through Kolmogorov-Smirnov testing underlines the statistical congruence between the spin period and spin period derivative distributions of both observations and simulations. Moreover simulations with a higher gamma-ray instrumental sensitivity show that the GeV excess in the Galactic centre could be of pulsar origin.