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High-energy astrophysical neutrino fluxes are, for many applications, modeled as simple power laws as a function of energy. While this is reasonable in the case of neutrino production in hadronuclear $pp$ sources, it typically does not capture the behavior in photohadronic $p\gamma$ sources: in that case, the neutrino spectrum depends on the properties of the target photons the cosmic rays collide with and on possible magnetic-field effects on the secondary pions and muons. We show that the neutrino production from known photohadronic sources can be reproduced by a thermal (black-body) target-photon spectrum if one suitably adjusts the temperature, thanks to multi-pion production processes. This allows discussing neutrino production from most known $p\gamma$ sources, such as gamma-ray bursts, active galactic nuclei and tidal disruption events, in terms of a few parameters. We apply this thermal model to study the sensitivity of different classes of neutrino telescopes to photohadronic sources: we classify the model parameter space according to which experiment is most suitable for detection of a specific source class and demonstrate that different experiment classes, such as dense arrays, conventional neutrino telescopes, or radio-detection experiments, cover different parts of the parameter space. Since the model can also reproduce the flavor and neutrino-antineutrino composition, we study the impact on the track-to-shower ratio and the Glashow resonance., Comment: 38 pages, 13 figures; data available at https://github.com/damianofiorillo/Unified-thermal-model