A turbine engine stator vane is provided that rotates about an axis, and includes an airfoil, a flange and a shaft. The airfoil extends axially between a first airfoil end and a second airfoil end. The airfoil includes a concave side surface, a convex side surface and a cavity. The concave and the convex side surfaces extend between an airfoil leading edge and an airfoil trailing edge. The cavity extends axially into the airfoil from a cavity inlet in an end surface at the second airfoil end. The flange is connected to the second airfoil end. The flange extends circumferentially around at least a portion of the cavity inlet, and radially away from the concave and the convex side surfaces to a distal flange edge. The shaft extends along the axis, and is connected to the second airfoil end.