"Since there is no way with an iPhone to date stamp photos, I wonder how good photographing the car really is push comes to shove."
EXIF data is shown on iPhone photos, however that may be dependent on which iOS version the phone is using. I'm currently using iOS 16.5.1 and with the iPhone camera app, tapping the small "i" that's in a circle below the photo will reveal not only time and date, but also the camera settings used to capture the image (ie: shutter speed, ISO, etc. That's also true of the Camera+ app that I use. Info is also provided on videos (ie: FPS, date, etc.). If I post an iPhone photo to my photo website, the information will also be shown there.
If the phone is set to update time automatically, it should adjust to European time as soon as the phone is switched on in Europe. The local time would then be attached to photos. I use an iPhone and that's always been the case when travelling.