As somebody who has used wireless CarPlay for a year now let me say this -
There are times it is nice, such as in the winter when you don't have to take your gloves off to fish it out under multiple layers of clothes - but most of the time in actual use you tend to want to plug it in anyway...
Sounds like something that can be fixed there. If it can do one it should be able to do the other. Unless you are playing music that is downloaded to your phone and not from the cloud?
Try sending a text! That's what originally made me remind you about using Siri - you had said you pulled over...
The one thing iOS 13 fixes that absolutely drives me insane in the current version - it will no longer change what's on the CarPlay screen when you use your phone. That was always terrible UI.
Another tip, regarding needing to back out of CarPlay to access radio -
If you use apps like iHeartRadio, TuneIn and Radio.com you can access stations through those instead. Better quality too.
I've had wireless CarPlay in my Hyundai Elantra GT since last summer. The wireless unit creates a wi-fi network for your car - that's how you are connecting to it.
FWIW - when I do connect via USB to charge my phone, I do not notice a change in audio quality.
This is the unit I use (came out...