We illustrate the extraordinary potential of the (far-IR) Origins Survey Spectrometer (OSS) on board the Origins Space Telescope (OST) to address a variety of open issues on the co-evolution of galaxies and AGNs. We present predictions for blind surveys, each of 1000 h, with different mapped areas (a shallow survey covering an area of 10 deg2 and a deep survey of 1 deg2) and two different concepts of the OST/OSS: with a 5.9 m telescope (Concept 2, our reference configuration) and with a 9.1 m telescope (Concept 1, previous configuration). In 1 000 h, surveys with the reference concept will detect from ∼1.9×106 to ∼8.7×106 lines from ∼4.8×105 to 2.7×106 star-forming galaxies and from ∼1.4×104 to ∼3.8×104 lines from ∼1.3×104 to 3.5×104 AGNs. The shallow survey will detect substantially more sources than the deep one; the advantage of the latter in pushing detections to lower luminosities/higher redshifts turns out to be quite limited. The OST/OSS will reach, in the same observing time, line fluxes more than one order of magnitude fainter than the SPICA/SMI and will cover a much broader redshift range. In particular it will detect tens of thousands of galaxies at z ≥ 5, beyond the reach of that instrument. The polycyclic aromatic hydrocarbons lines are potentially bright enough to allow the detection of hundreds of thousands of star-forming galaxies up to z ∼ 8.5, i.e. all the way through the reionisation epoch. The proposed surveys will allow us to explore the galaxy–AGN co-evolution up to z ∼ 5.5−6 with very good statistics. OST Concept 1 does not offer significant advantages for the scientific goals presented here.