I think it would happen this way: firstly you would automate things that go on a track - like trains - which I think we already have. Then secondly you could have certain lanes or roads that are designed for specific self-driving vehicles. This could help buses and trucks (in certain areas) get around. Then you would keep adding more of those, until many roads have those and very few manual lanes. It wouldn't immedia…
Look at Uber, Lyft, Bird, AirBnB, and many other companies who just implement whatever they feel like and dare cities to challenge them. The path of corporate-profit-seeking SDC's seems far more likely than your vision of limitation and safety.
In fact, if we actually look what's happening on the ground, Waymo is advanced in Arizona, a regulation-light state. Tesla is pushing advancing features, hardware, and software across markets, though there are some exceptions made. Still, these companies are going to push for autonomy-everywhere faster than most would deem comfortable.