The actin cortex of an animal cell is a thin polymeric layer attached to the inner side of the plasma membrane. It plays a key role in shape regulation and pattern formation on the cellular and tissue scale and, in particular, generates the contractile ring during cell division. Experimental studies showed that the cortex is fluid-like but highly viscous on long time scales with a mechanics that is sensitively regulated by active and passive cross-linker molecules that tune active stress and shear viscosity. Here, we use an established minimal model of active surface dynamics of the cell cortex supplemented with the experimentally motivated feature of mechanosensitivity in cross-linker binding dynamics. Performing linear stability analysis and computer simulations, we show that cross-linker mechanosensitivity significantly enhances the versatility of pattern formation and enables self-organized formation of contractile rings. Furthermore, we address the scenario of concentration-dependent shear viscosities as a way to stabilize ring-like patterns and constriction in the mid-plane of the active surface.