Detecting TeV–PeV cosmic neutrinos provides crucial tests of neutrino physics and astrophysics. The statistics of IceCube and the larger proposed IceCube-Gen2 demand calculations of neutrino-nucleus interactions subdominant to deep-inelastic scattering, which is mediated by weak-boson couplings to nuclei. The largest such interactions are $W$-boson and trident production, which are mediated instead through photon couplings to nuclei. In a companion paper [B. Zhou and J. F. Beacom, following Paper, Neutrino-nucleus cross sections for $W$-boson and trident production, Phys. Rev. D 101, 036011 (2020).], we make the most comprehensive and precise calculations of those interactions at high energies. In this paper, we study their phenomenological consequences. We find that: (1) These interactions are dominated by the production of on-shell $W$ bosons, which carry most of the neutrino energy; (2) the cross section on water/iron can be as large as $7.5\%/14\%$ that of charged-current deep-inelastic scattering, much larger than the quoted uncertainty on the latter, (3) attenuation in Earth is increased by as much as 15%; (4) $W$-boson production on nuclei exceeds that through the Glashow resonance on electrons by a factor of $\simeq 20$ for the best-fit IceCube spectrum; (5) the primary signals are showers that will significantly affect the detection rate in IceCube-Gen2; a small fraction of events give unique signatures that may be detected sooner.