My previous Droid would let me answer phone calls AND read and let me respond to my text messages to me via bluetooth connection in my 2016 Chevy Tahoe (MyLink). Since "upgrading" to the Moto Z Force Droid, I can still do the phone calls, but the text read/response does not work.
Of course. To be certain, prior or since the change in device, have you checked with the vehicle's manufacturer to determine if the new device would have the same compatibility with your vehicle and it's software/hardware?
Could be a problem with that system not supporting the newer device.
I linked Chevrolet's guide below, but for Motorola I only see Moto X 2nd Gen listed. They generally don't guarantee perfect compatibility with other devices, but you can reach out to the dealer directly to confirm.