Deciphering the conditions under which neutron captures occur in the Universe to synthesize heavy elements is an endeavor pursued since the 1950s, but has proved elusive up to now due to the experimental difficulty of generating the extreme neutron fluxes required. It has been evoked that laser-driven (pulsed) neutron sources could produce neutron beams with characteristics suitable to achieve nucleosynthesis in the laboratory. In this scheme, the laser first generates an ultra-high-current, high-energy proton beam, which is subsequently converted into a dense neutron beam. Here we model, in a self-consistent manner, the transport of laser-accelerated protons through the neutron converter, the subsequent neutron generation and propagation, and finally the neutron capture reactions in gold (${}^{197}\mathrm{Au}$), chosen as an illustrative example. Using the parameters of present-day available lasers, as well as of those foreseeable in the near future, we find that the final yield of the isotopes containing two more neutrons than the seed nuclei is negligible. Our investigation highlights that the areal density of the laser-driven neutron source is a critical quantity and that it would have to be increased by several orders of magnitude over the current state of the art in order to offer realistic prospects for laser-based generation of neutron-rich isotopes.