Using computer simulation on a model of oxygen ordering in the basal Cu_1-yM_yO_x plane of the high-temperature superconductor YBa₂Cu_3-yM_yO_6+x, we show that the effect of substituting Cu with M = Co, Fe, or Al is a diminishing of the oxygen-ordered orthorhombic domain sizes and eventually a breakdown of the orthorhombic structure. The model accounts for experimental structural data. By use of a previously established minimal model, which connects the superconducting transition temperature, T_c, to the formation of specific ordered orthorhombic domains in the undoped material, we show that the detrimental effect on T_c by metal-ion doping as well as oxygen depletion becomes manifest in a unified way as a result of oxygen disordering.