One of the long-term goals of exoplanet science is the atmospheric characterization of dozens of small exoplanets in order to understand their diversity and search for habitable worlds and potential biosignatures. Achieving this goal requires a space mission of sufficient scale. We seek to quantify the exoplanet detection performance of a space-based mid-infrared nulling interferometer that measures the thermal emission of exoplanets. For this, we have developed an instrument simulator that considers all major astrophysical noise sources and coupled it with Monte Carlo simulations of a synthetic exoplanet population around main-sequence stars within 20 pc. This allows us to quantify the number (and types) of exoplanets that our mission concept could detect over a certain time period. Two different scenarios to distribute the observing time among the stellar targets are discussed and different apertures sizes and wavelength ranges are considered. Within a 2.5-year initial search phase, an interferometer consisting of four 2 m apertures covering a wavelength range between 4 and 18.5 $mu$m could detect up to ~550 exoplanets with radii between 0.5 and 6 R$_oplus$ with an integrated SNR$ge$7. At least ~160 of the detected exoplanets have radii $le$1.5 R$_oplus$. Depending on the observing scenario, ~25-45 rocky exoplanets (objects with radii between 0.5 and 1.5 $_{oplus}$) orbiting within the empirical habitable zone (eHZ) of their host stars are among the detections. With four times 3.5 m aperture size, the total number of detections can increase to up to ~770, including ~60-80 rocky, eHZ planets. With four times 1 m aperture size, the maximum detection yield is ~315 exoplanets, including $le$20 rocky, eHZ planets. In terms of predicted detection yield, such a mission can compete with large single-aperture reflected light missions. (abridged)
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