IceCube-Gen2, the next generation of the IceCube Neutrino Observatory at the South Pole, will consist of three co-located arrays: a deep Optical Array and a more shallow and larger Radio Array for neutrino detection in the ice, and a Surface Array above the footprint of the Optical Array. The Surface Array will be comprised of hybrid stations featuring elevated radio antennas and scintillation detectors, following the design of a prototype station successfully operating at the South Pole since 2020. Besides providing a veto for neutrino detection, the Surface Array will make IceCube-Gen2 a unique laboratory for cosmic-ray air showers. Compared to the current IceCube detector with its IceTop surface array, the aperture for coincident air-shower measurements detected by both, the deep optical and surface arrays, will increase by about a factor of 30. In addition to particle physics questions, such as the production of PeV muons and neutrinos in prompt decays, these surface-deep coincidences will be used to target astrophysical questions of the most energetic Galactic cosmic rays. The combination of particle and radio measurements at the surface and high-energy muons measured in the ice promises unprecedented accuracy for the mass composition in the energy range of the presumed Galactic-to-extragalactic transition – complementing the multimessenger science case of IceCube-Gen2. This proceeding provides an overview of the IceCube-Gen2 Surface Array and, in particular, its radio component.