With their unique structural, optical and electrical properties, III-V nanowires (NWs) are an extremely attractive option for the direct growth of III-Vs on Si for tandem solar cell applications. Here, we introduce a core-shell GaAs/GaInP NW solar cell grown by molecular beam epitaxy on a patterned Si substrate, and we present an in-depth investigation of its optoelectronic properties and limitations. We report a power conversion efficiency of almost 3.7%, and a state-of-the-art open-circuit voltage (VOC) for a NW array solar cell on Si of 0.65 V. We also present the first quantification of the quasi-Fermi level splitting in NW array solar cells using hyperspectral photoluminescence measurements. A value of 0.84 eV is obtained at 1 sun (1.01 eV at 81 suns), which is significantly higher than qVOC. It indicates NWs with a better intrinsic optoelectronic quality than what could be expected from TEM images or deduced from electrical measurements. Optical and electronic simulations provide insights into the main absorption and electrical losses, and guidelines to design and fabricate higher-efficiency devices. It suggests that improvements at the n-type contact (GaInP/ITO) are key to unlocking the potential of next generation NW solar cells.