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Context. Many physical processes taking place during the evolution of binary stellar systems remain poorly understood. The ever-expanding observational sample of X-ray binaries (XRBs) makes them excellent laboratories for constraining binary evolution theory. Such constraints and useful insights can be obtained by studying the effects of various physical assumptions on synthetic X-ray luminosity functions (XLFs) and comparing them with observed XLFs. Aims. In this work we focus on high-mass X-ray binaries (HMXBs) and study the effects on the XLF of various, poorly constrained assumptions regarding physical processes, such as the common-envelope phase, core collapse, and wind-fed accretion. Methods. We used the new binary population synthesis code POSYDON , which employs extensive precomputed grids of detailed stellar structure and binary evolution models, to simulate the entire evolution of binaries. We generated 96 synthetic XRB populations corresponding to different combinations of model assumptions, including different prescriptions for supernova kicks, supernova remnant masses, common-envelope evolution, circularization at the onset of Roche-lobe overflow, and observable wind-fed accretion. Results. The generated HMXB XLFs are feature-rich, deviating from the commonly assumed single power law. We find a break in our synthetic XLF at luminosity ∼10 38 erg s −1 , similar to observed XLFs. However, we also find a general overabundance of XRBs (up to a factor of ∼10 for certain model parameter combinations) driven primarily by XRBs with black hole accretors. Assumptions about the transient behavior of Be XRBs, asymmetric supernova kicks, and common-envelope physics can significantly affect the shape and normalization of our synthetic XLFs. We find that less well-studied assumptions regarding the circularization of the orbit at the onset of Roche-lobe overflow and criteria for the formation of an X-ray-emitting accretion disk around wind-accreting black holes can also impact our synthetic XLFs and reduce the discrepancy with observations. Conclusions. Our synthetic XLFs do not always agree well with observations, especially at intermediate X-ray luminosities, which is likely due to uncertainties in the adopted physical assumptions. While some model parameters leave distinct imprints on the shape of the synthetic XLFs and can reduce this deviation, others do not have a significant effect overall. Our study reveals the importance of large-scale parameter studies, highlighting the power of XRBs in constraining binary evolution theory.