Depending on your definition of "internet connected," I'd disagree with your first point. Most new cars come equipped with the hardware for a data-enabled cellular connection and many have been for years. While it's not a feature you necessarily see or know about, it's there. Presumably, the ones that are used by a service like OnStar are constantly polling in the background to get status updates for the account the vehicle is associated with.

But yes, to my knowledge, most (all?) operate fine without a cell signal.

I have heard of various ride share services stranding folks in the middle of nowhere because of lack of signal.

Not heard a single story of an internet connected car not having basic functionality work without internet.

There was a story about a rental car stopping when it crossed a border, because the rental package was domestic use only.