Maybe simple envelopes would be more accurate. Any waveform played with a rectangular envelope will sound "organy." Add modulation and a more nuanced envelope to avoid this.
Yes, electronic organs often have rectangular envelopes. Pipe speech has an envelope that is far more complex than one might expect; indeed, one of the goals of voicing is to impart an envelope that allows the pipe to be "articulate", to a degree dependent on the caracteristics of the stop of which it is a member and depending on the overall tonal milieu of the specific instrument. Taken to its extreme, this results in a clunky "chiff" sound at the start of each note, something often heard in instruments designed in the "American Classic" and "Neo-Baroque" styles; but with more refinement, it can produce the gentle puff at the beginning of a flute note, or the "bubble" articulation of a French Horn. And at the other end of the envelope is a characteristic decay, initiated by the closing of the valve beneath the pipe and sustained by the resonance of the pipe body (which, in large pipes, can go on for several seconds).
I didn't know the term "chiff," but looking up videos of the sound, it's very similar to an effect I accidentally created in a recent digital synthesis project by using a low sample rate for the envelope (2kHz). I think the stair-stepped shape of the attack created some inharmonics in the audio.
Is chiff the result of inharmonics created by brief turbulence in the airflow?
Yes indeed. In extreme cases, it can give the impression of a xylophone! To control chiff, we use a pointy double-sided knife called a "nicking tool" to create a series of evenly-spaced nicks across the entire width of the flue (which is the slot at the base of the mouth, formed by the lower lip and the languid inside the pipe; air leaving the foot of the pipe is focused by the flue on its way towards the upper lip). Often the nicks are cut into both the lower lip and the languid at similar depths and perpendicular to the flue, although there are exceptions; the width, depth, and spacing of the nicks all have tonal implications. Care must be taken to avoid over-nicking, as the articulation -- some of which is necessary even when chiffing is not desired, and which may need to be more prominent when close to the pipe than when listening from the room -- can be completely lost.