We don't know that there is a problem on Apple's end despite your comments; others with Lexus CarPlay systems elsewhere have had no issues, and of those that have, either deleting all known devices and reassociating or using a known good USB-C to USB-A cable rated for data have solved many problems.
I trust you've tried both, so I would suggest working directly with Apple Support if you are not already:
Contact - Official Apple Support
Also be sure to contact your Lexus dealer and make sure your vehicle is running the most up to date version of firmware for your entertainment system that is available.