Well that is a matter of semantics. You are right in that the shutter speed does not affect the flash output, but a slower speed does force a smaller aperture which in turn reduces the fill. So I guess it is safer to say it reduces the amount of recorded fill, which is the same thing.
You can control fill by changing either the flash's power output or aperture. I don't perceive fill as being controlled by output alone. I view fill as the percentage of exposure contributed by the flash's light.
Yes the shutter speed allows more ambient light by 1/3 stop, but the reduced aperture reduces the ambient light by 1/3 stop to maintain ambient exposure. The difference is that less flash is recorded, when the same amount of ambient light is recorded.
If you had a copal shutter you could speed the shutter up and open the aperture up by the same number of stops, maintaining ambient and getting "more fill" by recording more of the flash. Aperture governs flash exposure, not shutter speed, because, as you said, the flash is to brief to be governed by shutter speed. Fill is governed by aperture.
Scott Chapin Powder Springs, GA, USA Nikonians Team Member