I think it is more likely that our cars (and all cars that cannot communicate with other collision-avoiding, computer-controlled/autonomous vehicles) will be prohibited from driving on most highways/thoroughfares, citing safety reasons, much the same way you can see signs at entrance ramps of highways today listing the prohibited means of transportation such as bicycles, horses, motorized bicycles, pedestrians, etc. If you want to drive on roads that are built or maintained using federal funds, you may have to use an approved vehicle or have your current vehicle retrofitted (at your own expense) to comply with the requirement to be able to coexist with compliant vehicles.
I foresee a time when antique cars will have to be trailered to privately owned driving courses, much the same way people bring their horses in trailers to go trail riding on private lands today, or the way certain non-compliant sports cars/race cars are taken to Lime Rock Park in CT, or drag strips across the country. Exceptions might be made as there are today for horses in Holiday parades like Memorial Day or Independence Day moving down Main St. USA, or the fancy go-karts the Shriners pilot, or the decorated bicycles that the kids ride, etc.