Ok, I have been tracking this forum and following this since we purchased a 2022 Hyundai Palisades with Wireless Carplay. At the time of purchase, I was using AT&T with an iPhone XR. We experienced all of the connectivity and "dropping" problems documented here. It was very frustrating as the wireless option was one of the benefits of buying a new vehicle.
Since then, I have upgraded my device to an iPhone 13 (5G) and more importantly, I switched to T-Mobile who - in my area - has stronger 5G coverage. After the switch, we have experienced ZERO drops with Apple Car Play. I went from constant drops... to ZERO drops. You can read that as you want, but this points to the problem being related to SIGNAL speed/strength from the wireless network to your device.
What I don't understand is why the device, which is locally connected to the car via bluetooth and local wifi drops screen projection when the signal to the device is poor (network). It shouldn't matter. Yes, the streaming content should pause (Streaming audio for Example) but the connection from the device to the car shouldn't be impacted by this.
I am not an engineer, so I don't have a fix. I can only explain my experience. This issue hasn't come up for me in 2 months. After I swapped carriers and moved to 5G, it just worked. Even when I don't have 5G coverage - which happens for sure as 5G coverage doesn't blanket my area yet.
So, look harder at your devices and your carriers. As mentioned above I still don't understand why signal strength impacts the connection to the car... but it seems to...