Whenever you take a flash photo, you're basically combining two exposures together in one shot: the flash exposure made with light from the flash, and ambient exposure, made with all the light that's not from the flash. And the controls for those two types of exposure are different.
Ambient exposure is controlled by iso, aperture, and shutter speed.
Flash exposure is controlled by iso, aperture, power, and distance.
And those differences in control mean you can balance the flash against the aperture however you want (within gear limits).
Now, several different things are going on here.
I shot in Av with ISO 400, then 200, 1/200 sec exposure and f/5.0. ETTL was enabled. I used exposure compensation to darken (-2EV) the images so they would be acceptable.
The first issue here is that if you were in sunny-16 conditions, then ISO 200, 1/200s exposure and f/5.0 puts you at more than +3EV overexposure just in the ambient. Adding anything with the flash would push that overexposure even farther. How you got these settings with -2EV exposure compensation I don't quite understand.
…Also, my camera would not let me select a higher shutter speed in Av mode.
The chances are very good that your initial settings were -2EV, but then you got overexposed by not having the flash set to use HSS (high-speed sync), and you were then limited to the camera's sync speed of 1/200s. High-speed sync must be turned on to use faster shutter speeds, otherwise you'll get dark/black bands with flash. (See Neil van Niekerk's HSS tutorial for more information).
I wanted a pretty opened f-stop to blur the backgrounds, hence f:5.0.
The problem is that if you're stuck at your lowest ISO setting, with a bigger aperture setting, and you're shooting in sunlight outdoors (even on a cloudy day), you may need to shoot with a shutter speed faster than 1/200s to avoid overexposure in the ambient, never mind getting -2EV below in the ambient.
So, you can use HSS to get there. But HSS will probably cost you -2EV in power output over not-using HSS. So you may also want to consider (if your subject isn't moving, and 1/200s is okay, speedwise) using an neutral density (ND) filter on your lens to act like sunglasses and bring the ambient exposure down that way.