The application of a gate voltage to control the superconducting current flowing through a nanoscale superconducting constriction, named as gate-controlled supercurrent (GCS), has raised great interest for fundamental and technological reasons. To gain a deeper understanding of this effect and develop superconducting technologies based on it, the material and physical parameters crucial for the GCS effect must be identified. Top-down fabrication protocols should also be optimized to increase device scalability, although studies suggest that top-down fabricated devices are more resilient to show a GCS. Here, we investigate gated superconducting nanobridges made with a top-down fabrication process from thin films of the non-centrosymmetric superconductor niobium rhenium with varying ratios of the constituents (NbRe). Unlike other devices previously reported and made with a top-down approach, our NbRe devices systematically exhibit a GCS effect when they were fabricated from NbRe thin films with small grain size and etched in specific conditions. These observations pave the way for the realization of top-down-made GCS devices with high scalability. Our results also imply that physical parameters like structural disorder and surface physical properties of the nanobridges, which can be in turn modified by the fabrication process, are crucial for a GCS observation, providing therefore also important insights into the physics underlying the GCS effect.