The nature of the gas in CO-rich debris discs remains poorly understood, as it could either be a remnant from the earlier Class II phase or of secondary origin, driven by the destruction of icy planetesimals. The aim of this paper was to elucidate the origin of the gas content in the debris discs via various simple molecules that are often detected in the less-evolved Class II discs. We present millimetre molecular line observations of nine circumstellar discs around A-type stars: four CO-rich debris discs (HD 21997, HD 121617, HD 131488, HD 131835) and five old Herbig Ae protoplanetary discs (HD 139614, HD 141569, HD 142666, HD 145718, HD 100453). The sources were observed with the Atacama Large Millimeter/submillimeter Array (ALMA) in Bands 5 and 6 with 1-2 arcsec resolution. The Herbig Ae discs are detected in the CO isotopologues, CN, HCN, HCO<SUP>+</SUP>, C<SUB>2</SUB>H, and CS lines. In contrast, only CO isotopologues are detected in the debris discs, showing a similar amount of CO to that found in the Herbig Ae protoplanetary discs. Using chemical and radiative transfer modelling, we show that the abundances of molecules other than CO in debris discs are expected to be very low. We consider multiple sets of initial elemental abundances with various degrees of H<SUB>2</SUB> depletion. We find that the HCO<SUP>+</SUP> lines should be the second brightest after the CO lines, and that their intensities strongly depend on the overall CO/H<SUB>2</SUB> ratio of the gas. However, even in the ISM-like scenario, the simulated HCO<SUP>+</SUP> emission remains weak as required by our non-detections.< Réduire