The cosmological recombination radiation (CRR) is one of the inevitable Lambda cold dark matter spectral distortions of the cosmic microwave background (CMB). While it shows a rich spectral structure across dm-mm wavelengths, it is also one of the smallest signals to target. Here, we carry out a detailed forecast for the expected sensitivity levels required to not only detect but also extract cosmological information from the CRR in the presence of foregrounds. We use CosmoSpec to compute the CRR including all important radiative transfer effects and modifications to the recombination dynamics. We confirm that detections of the overall CRR signal are possible with spectrometer concepts like SuperPIXIE. However, for a real exploitation of the cosmological information, an ≃ 50 times more sensitive spectrometer is required. While extremely futuristic, this could provide independent constraints on the primordial helium abundance, Yp, and probe the presence of extra relativistic degrees of freedom during BBN and recombination. Significantly improving the constraints on other cosmological parameters requires even higher sensitivity (another factor of ≃5) when considering a combination of a CMB spectrometer with existing CMB data. To a large part, this is due to astrophysical foregrounds which interestingly do not degrade the constraints on Yp and Neff as much. A future CMB spectrometer could thus open a novel way of probing non-standard BBN scenarios, dark radiation and sterile neutrinos. In addition, inflation physics could be indirectly probed using the CRR in combination with existing and forthcoming CMB anisotropy data.