I saw an interactive tutorial at http://www.sekonic.com/classroom/fillflashtutorial.aspx,, it talks about how to control the proportion of fill flash relative to ambient light. In it, they illustrate how by changing aperture you can change this flash/ambient light ratio. But I have always believed that you need to change shutter speed instead. The faster the speed(up the maximum synch speed)the less ambient light that gets into the picture. They claim that more you open the aperture, the less flash light percentage you get in your pictures.
Neither the methods used by the tutorial nor those used by your previous understanding are wrong, they are just based on different starting points and sets of assumptions. Perhaps your confusion stems from an incomplete understanding of both methods. A little knowledge can sometimes be a dangerous thing. :-)
When using manually controlled flash you have three on-camera variables that affect exposure and the flash-to-ambient ratio: ISO, Aperture, and Shutter Speed. Assuming the ambient light is a constant brightness and the subject has a constant amount of reflectivity, the fourth and only other variable is not on-camera. It is Flash Power.
For the sake of this discussion, let's leave ISO constant throughout. Of the other three variables, which method you use depends on which two variable are subject to being adjusted and which one you desire to remain constant.
The tutorial takes the approach that shutter speed is constant and the two variables in play are flash power and aperture. The flash to ambient light ratio is determined by changing the power of the flash. The prime concern when setting the aperture is maintaining proper exposure of the subject and is made without a corresponding change in shutter speed. In other words, you offset the increase in flash power by decreasing the size of the aperture and make no change to the shutter speed. The reason the aperture can be reduced without changing the overall exposure of the subject is because the flash power is also being increased by the same amount. For each stop the aperture is narrowed, the flash power is increased by one stop. (Either that or the flash is being moved closer to the subject to have the same effect. Remember at twice the distance a flash of a specific power only provides one-fourth the illumination.) Since the shutter speed isn't also being adjusted to compensate for the narrower aperture, the influence of the ambient light is reduced and the background is darker.
The second half of your question that expresses what you have always held to be true is also correct. This approach is based on the assumption that the the amount of flash power is constant and the two variables in play are only shutter speed and aperture. Without a corresponding change in flash power, aperture affects the exposure of the subject caused by both flash and ambient light as well as the exposure of the background caused by the ambient light. Shutter then affects the exposure of only the ambient light on both the subject and the background. If you decrease the amount of exposure created by the flash by narrowing the aperture, then you must also lengthen the shutter time to allow the ambient light to provide enough additional exposure to maintain the same overall exposure of your subject. The longer exposure time also causes the background to be brighter in the resulting photo.
You could also take a third approach: Make the aperture the constant, adjust the shutter speed until the background reaches the desired exposure and then adjust the flash power to properly expose your subject.
Notice that step 2 in their process is:
So before you even touch your camera, you're adjusting the ratio of flash to ambient by changing the power and/or position of the flash and measuring with the meter (remember, this is a tutorial on how to use a flash meter). It's only when the desired ratio has been achieved that you pick up your camera, apply the settings from the meter, and shoot.
I think you're misunderstanding what they're saying. You're right that you can use aperture to control flash and shutter speed to control ambient. In this case, though, the flash/ambient ratio has already been determined, and you're simply adjusting the camera for proper exposure.
For example, let's say that you've taken a photo that's properly exposed, but you want more flash and less ambient. One way to do that is to bump up the power on the strobe, but that would result in an overexposed photo because the total amount of light is now greater. To compensate, you use a smaller aperture. You can't use a faster shutter speed and maintain the flash/ambient ratio because, as you correctly point out, shutter speed only affects ambient. (Also, you can't increase shutter beyond the flash sync speed without getting into high speed sync, and that affects the available flash power.) You could decrease the ISO setting, but the ISO in the example is already down to 160. Changing the aperture decreases the total light hitting the sensor without affecting the flash/ambient ratio.
Indirectly, letting in more light will allow for a faster shutter speed and thus the flash will be a higher percentage of the light, however if the shutter speed isn't changed, you are correct that the aperture would let in more light from both the ambient and the flash since the flash is only different from the ambient in the length of time that it is active.