I was talking about stock Android on Pixels, as the question was about updating to Android 13 and a Pixel 6 Pro.Taken just now from my wife’s phone:
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Though this is Android 11 so not sure if it’s been recently deprecated. After turning off optimisation my wife’s phone has never had an issue with acting as a phone key and waking the car.
I'm guessing your wife has some Chinese or possibly Samsung phone.
Speaking with my Android developer hat on, the Chinese manufacturers are a blot on the landscape, as they insist on killing apps in the background and doing all sorts of stupid things to running apps. So steer clear of Oppo, Xiaomi and definitely not Huawei as they have lost Google support.