That saying that "aperture controls flash and shutter controls ambient" is only somewhat a little bit true in the context of flash pictures. Aperture always affects BOTH flash and ambient equally. However, shutter speed does not affect fast flash, therefore flash users can think of shutter speed as favoring ambient, and it can be a helpful concept that way, in a limited way.
Specifically, using a slow shutter maximizes effect of continuous ambient, and a fast shutter speed minimizes it, without affecting flash. So that choice can affect the flash to ambient ratio seen in the picture, as a photographers choice. Shutter speed DOES affect that ratio. Since flash power level and the aperture setting are related, and since the shutter speed does not affect the flash exposure, it might seem as if the statement had some merit. And it can be a useful guide, even if not technically true.
But very absolutely, aperture affects all light, including ambient and/or flash, equally. Same for ISO, affects all light, equally. Shutter speed just happens not to affect flash exposure (since the flash is faster than the shutter, it does not matter how much longer the shutter might remain open, since the shorter flash already finished, and remains the same).
For a black background, try to keep any and all significant light off of it. One way is that a fast shutter speed (maximum sync speed) will limit the ambient, including that on the background (and the flashes don't have to be aimed at the background). Even white can be made to appear black (without light on it), but starting with a black background is easier.