Why does a metal flute not sound substantially different from a wooden flute for that reason alone?

A metal flute does not sound substantially different from a wooden flute because the primary source of sound in a flute is the vibration of the column of air contained within the instrument, rather than the vibration of the instrument's body material itself.1

While the material of an instrument can influence the sound of percussion instruments, the characteristic timbre of a wind instrument is primarily determined by the length and shape of its tube. These physical dimensions dictate the presence and intensity of overtones, which provide the instrument with its unique color and quality. Because the air column is the vibrating medium, the specific material used to construct the tube has a negligible effect on the resulting sound compared to the instrument's geometry.12