The iPhone's volume not changing is a slightly different issue from whether iOS is digitally amplifying the signal (which would be evidenced by distortion when over-amplified). I realize you already know this, I'm just pointing it out for others who want to test their devices.
The issue with your wife's iPhone is probably more related to an issue I've encountered where sometimes the iPhone's individual sound sliders don't present themselves to the user properly. Android also flubs this but the screen/drawer/settings usually present all of the sliders to the user so it can be addressed quickly. But on an iPhone the user almost always only has access to one slider at a time depending on what the device believes it's connected to and the other sliders are nested deep in the main settings.
It's possible (in my experience, likely) that her iPhone isn't recognizing that adapter for what it is and so when she's using the volume rocker it's only changing the phone volume (and not changing speaker volume or bluetooth volume, etc.). Try changing the volume from inside the settings menu making sure you're controlling the bluetooth slider and see if it changes anything in her car.
FWIW, I have my in-car settings as BT volume at max, phone BT at max, EQ 0, while controlling the sound from the stereo and I don't detect any distortion.