Multi-conjugated adaptive optics (MCAO) is essential for performing astrometry with the Extremely Large Telescope (ELT). Unlike most of the 8-m class telescopes, the ELT will be a fully adaptive telescope, and a significant portion of the adaptive optics (AO) dynamic range will be depleted by the correction and stabilization of the telescope aberrations and instabilities. MCAO systems are of particular interest for ground-based astrometry since they stabilize the low-order field distortions and transient plate scale instabilities, which originate from the telescope and in the instrument. All instruments have several optical elements relatively far away from the pupil that can potentially challenge the astrometric precision of the observations with their residual mid-spatial frequencies errors. Using a combined simulation of ray tracing and AO numerical codes, we assess the impact of these systematic errors at different field-of-view (FoV) scales and fitting scenarios. The distortions have been assessed at different sky position angles (PA) and indicate that over large FoVs only small PA ranges (±1 deg to 3 deg) are accessible with astrometric residuals ≤50 μas. A full compliance with the astrometric requirement, at any PA, is achievable for 2 arc sec2 FoV patches already with a third-order polynomial. The natural partition of the optical system into three segments, i.e., the ELT, the MAORY MCAO module, and the MICADO instrument, resembles a splitting of the astrometric problem into the three subsystems that are characterized by different distortion amplitudes and calibration strategies. The result is a family portrait of the different optical segments with their specifications, dynamic motions, conjugation height, and AO correctability, leading to tracing their role in the bigger puzzle of the 50-μas as astrometric endeavor.