Upcoming neutrino telescopes may discover ultra-high-energy (UHE) cosmic neutrinos, with energies beyond 100 PeV, in the next 10-20 years. Finding their sources would identify guaranteed sites of interaction of UHE cosmic rays, whose origin is unknown. We search for sources by looking for multiplets of UHE neutrinos arriving from similar directions. Our forecasts are state-of-the-art, geared at neutrino radio-detection in IceCube-Gen2. They account for detector energy and angular response, and for critical, but uncertain backgrounds. Sources at declination of -45 degrees to 0 degrees will be easiest to discover. Discovering even one steady-state source in 10 years would imply that the source has an UHE neutrino luminosity at least larger than about 1043 erg/s (depending on the source redshift evolution). Discovering no transient source would disfavor transient sources brighter than 1053 erg as dominant. Our results aim to inform the design of upcoming detectors.