Yes, flash usages and aperture are independent of each other. I shoot weddings and 90-95% of my flash usage is with wide open or near wide open apertures.
What is confusing you is specifically the case of shooting outdoors where you want the flash to fill in with power equivalent to daylight. In that situation, the shorter your shutter speed, the less daylight is let in and the more impact your flash has on the scene relative to the sun.
The problem is that cameras have something known as the maximum sync speed. This is the fastest shutter speed for which there exists a time when the shutter is fully open. A DSLR shutter is composed of two curtains, first one curtain opens and then the other curtain closes. When you are shooting a slower shutter speed than the sync speed, the first shutter opens fully before the second shutter begins closing. The flash can go off any time in this window (normally either right after the first curtain opens or right before the second curtain closes).
When shooting faster than the sync speed, the second shutter begins closing before the first shutter finishes opening. This means that there is no moment in time where the shutter can go off and fully expose the image. To avoid a dark band, HSS or high speed sync has to be used. High speed sync works like a strobe and produces rapid lower power flashes over the duration of the shutter. Since the flash has to discharge multiple times however, it can not discharge as much power overall, and thus some overall flash power is lost.
Making adjustments to aperture or adding ND filters won't make a difference on the impact of sunlight vs flash though as both also limit the amount of flash power that reaches the sensor as well, so it makes no difference at all on aperture selection. You select aperture for the desired depth of field and choose flash power and shutter speed based on the desired lighting.