Notice that step 2 in their process is:

Adjust the flash power output until you have a desired percentage of flash

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.

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.

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.