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I suspect it is on Tesla to update their interface to messages. Apple made significant changes to the way messages behaves, such as being able to edit a message after it is sent. I doubt Tesla expected this and will have to adapt their client. For me, I never liked the way messages behaved in the car, so I do not sync messages to/from my phone. It is easier just to watch messages on my Apple Watch. Mostly just a quick glance at my wrist. Phone contacts, frequents and recents are fine, but I skip messages and emails.
I’m on iOS 15 and Sync Messages stopped working recently. I have the Bluetooth setting enabled on my iPhone to show notifications but the Tesla software refuses to enable the option.