>Those tests don't account for our ability to sense the presence/absence of overtones.

Overtones are frequencies. If you filter the higher frequencies you remove the higher overtones. The only way you're going to "hear" ultrasound is if it's very loud and you hear audio-frequency distortion generated in your own ear, e.g. like when bats are squeaking nearby. But this isn't useful musically, because everybody's ears distort differently.