No, the eSIM is just working fine, as I described above. I can use it as a hotspot for my phone without a problem. But the car uses the eSIM onboard to make the connection to native apps like weather, alexa and Spotify. So if that's not available, the car has some kind of problem using the eSIM connection. I had it a few times that weather was not available (the rest I dont use), usually for an hour and the iPhone app still worked. It's new for me now that the car is not connected to the cloud so that all iPhone app features such as it's pre-conditioner / auxiliary heating, Secure Tracker, etc. are not available anymore.
And yes, every other day I have a problem with CarPlay also. But the phone's connection should not have anything to do with the car's ability to connect its own apps to the internet, because that's not passing through the phone unless you give the car a hotspot with your phone. 401 btw should be a no permission error.