Source of this information? I seriously doubt that it's "constantly pushing that image back and forth hundreds of times". Where did you get that idea?
Wireless CarPlay doesn't compress the music, it's not using Bluetooth to stream, it's using 5gHz WiFi.
You don't know what a display refresh rate is? I didnt provide a source because I figured it was common logic among car guys. (See I can be an asshole too ?)
So when casting an image wirelessly, to keep up with that refresh rate of display, the device has to send the image to the display through the air at minimum equal to the refresh rate of the display. Otherwise the display will look choppy/stuttering.
Most displays are 60Hz, but it's possible the display in the 2016-17 GTI's are only 30hz. That means it updates the image on the display 30 times per
second. The device casting that image has to send an updated image to cast to the display every one of those times.
So actually at 30 times at second it sends the image to the display through the air 1,800 times in just 1 minute. You still want a source? Just simply google "chromecast refresh rate" pulls up....
"Chromecast Ultra is designed to pick the maximum refresh rate (up to 60Hz) and resolution supported by your 4K Ultra HD TV's HDMI connection."
Source right here.
Chromecast uses the same technology as wireless Android Auto and wireless CarPlay. It casts the image over WiFi to your router, then from the router to the display. It does that 60 times per second. So to get an idea of how much it drains your battery, get a Chromecast and play it for 10 minutes from your phone to see how much battery drains. If using navigation at the same time, then it will drain even much faster than that.
I never said CarPlay compresses music. Even if it doesn't, then it's still not going to be nearly the same quality over the air, even though it's using WiFi instead of Bluetooth. It will be better than Bluetooth, but still not the same as a direct wired connection by auxiliary or by USB.
In regards to Bluetooth Audio quality vs. Aux or USB Audi quality, this is also another common known fact among not just audiophiles but anyone who reads about tech reviews.
Source right here.
It talks a ton about how Bluetooth Audio basically sucks. It does say how casting audio is much better over WiFi, but also talks about how casting audio over WiFi can still be a bottleneck, but just not as much as Bluetooth.