Context. Without a doubt, mass transfer in close binary systems contributes to the populations ofWolf-Rayet (WR) stars in the MilkyWay and the Magellanic Clouds. However, the binary formation channel is so far not well explored. Aims. We want to remedy this by exploring large grids of detailed binary and single star evolution models computed with the publicly available MESA code, for a metallicity appropriate for the Large Magellanic Cloud (LMC). Methods. The binary models were calculated through Roche-lobe overflow and mass transfer, until the initially more massive star exhausted helium in its core. We distinguish models of WR and helium stars based on the estimated stellar wind optical depth. We used these models to build a synthetic WR population, assuming constant star formation. Results. Our models can reproduce the WR population of the LMC to significant detail, including the number and luminosity functions of the main WR subtypes. We find that for binary fractions of 100% (50%), all LMC WR stars below 10⁶ L_⊙ (10^5.7 L_⊙) are stripped binary mass donors. We also identify several insightful mismatches. With a single star fraction of 50%, our models produce too many yellow supergiants, calling either for a larger initial binary fraction, or for enhanced mass loss near the Humphreys-Davidson limit. Our models predict more long-period WR binaries than observed, arguably due to an observational bias toward short periods. Our models also underpredict the shortest-period WR binaries, which may have implications for understanding the progenitors of double black hole mergers. Conclusions. The fraction of binary-produced WR stars may be larger than often assumed and outline the risk to miscalibrate stellar physics when only single star models are used to reproduce the observed WR stars.