The upcoming class of extremely large telescopes (ELTs) will provide an unprecedented opportunity to use high-resolution spectroscopy to characterize terrestrial exoplanets for habitability and life. In particular, these telescopes are likely the best near-term tools for detecting molecular oxygen in nearby exoplanet atmospheres. However, determining whether oxygen is more likely to have a biological origin requires contextual information from the planetary environment to support the identification and rule out false positives. Studies which investigate the ELTs’ capacity to detect other gases can enhance the science return from these telescopes and expand our abilities to search for signs of life. We have developed a novel pipeline to simulate telescope observations and estimate the detectability of a suite of gases—CH$_4$, CO$_2$, CO, O$_3$, and H$_2$O—that can help give context to ELT O$_2$ detections in terrestrial exoplanet atmospheres. As input, we used a suite of photochemically self-consistent simulations of M dwarf planets with modern/Archean Earth-like atmospheres, and worlds with abiotic O$_2$ buildup due to photochemical generation and ocean loss processes. We find that CO$_2$ and CH$_4$ are two of the most detectable molecules in M dwarf planetary atmospheres, and may be detectable on TRAPPIST-1 e in less than 35 transits. This may be the only known biosignature pair accessible with ELT high-resolution spectroscopy for this target. However, for closer targets, the ELTs alone may be capable of discriminating an inhabited world from one without life with tens of hours of observation time under ideal conditions. Additionally, we find the detectability of all gases is strongly dependent on host star type—planets orbiting late-type M dwarfs may require less overall observation time to achieve a significant detection. Finally, we develop an observing protocol that prioritizes the most detectable gas absorption bands to maximize the science output of ELT observations, and inform instrument development beyond the first light capabilities.