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The cosmological recombination radiation (CRR) is one of the inevitable $Λ$CDM 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 ${\tt CosmoSpec}$ 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 ${\it SuperPIXIE}$. However, for real exploitation of the cosmological information, a $\simeq 50$ times more sensitive spectrometer is required. While extremely futuristic, this could provide independent constraints on the primordial helium abundance, $Y_p$, 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 $\simeq 5$) 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 $Y_p$ and $N_{\rm eff}$ 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.