The discovery of ultra-high-energy (UHE) neutrinos has the potential to offer unique insight into fundamental questions. To capitalize on the upcoming opportunity provided by new UHE neutrino telescopes, we provide state-of-the-art forecasts of the discovery of a diffuse flux of UHE neutrinos over the next $10-20$ years, focusing on neutrino radio-detection in the planned IceCube-Gen2 detector. We use state-of-the-art flux predictions and detector modeling. We find that, even under conservative analysis choices, most benchmark UHE neutrino flux models from the literature may be discovered within 10 years of detector exposure, with many discoverable sooner, and may be distinguished from each other. Our results demonstrate the transformative potential of next-generation UHE neutrino telescopes.