Another thing you can try, turn off your BlueTooth on your phone, then try connecting to the adapter, by selecting the wireless CarPlay option, in Settings > General > CarPlay. Can you connect that way?
I kinda agree, that if your head unit won't display the firmware's connection screen, there is most likely a fundamental issue between the adapter and your Mazda head unit. As, even if my phone doesn't want to connect, bringing up "CarPlay" from my head unit, always show the Carlinkit/CPlay2Air's connection interface. If you're not getting that, it seems you head unit is being finicky.
I can only assume, that wired, everything works peachy? And you are using the same USB jack (in your car) that supports wired CarPlay? Sorry if these questions seem elementary, just ruling out the obvious.