A manufacturing method is provided during which a preform component for a turbine engine is provided. A cooling aperture is formed in the preform component. The cooling aperture includes a centerline, an inlet and an outlet. The cooling aperture extends longitudinally along the centerline through a wall of the preform component from the inlet to the outlet. The forming of the cooling aperture includes forming a first portion of the cooling aperture using a machining tool implement with a first toolpath that is angularly offset from the centerline by a first angle between thirty-five degrees and ninety degrees.