Why doesn't a sudden loud noise sound high pitched?
Pitch is a human psychoperceptual phenomena, which does not work the same way as a Fourier transform.

IIRC: Even if an FFT shows strong high frequency components in a short impulse or burst, the human ear-brain-system needs to detect roughly somewhere on the order of around 6 repetitions/periods/cycles of a periodic or pseudo-periodic acoustic waveform before communicating "I hear a pitch" to consciousness.  (depending somewhat on, but nowhere near linearly with, frequency). A single or duple period of even a pure sinewave won't do it.