I think he's referring to the idea that you'll use an Uber-like app (either Uber or if Tesla builds their own then the Tesla version) to order an EV to use it's self driving AI to come pick you up, take you to your destination, then drive off to service other customers.

https://newsroom.uber.com/pittsburgh-self-driving-uber/

I think you're right, but that change in behavior will come (if it comes at all) with self-driving technology, which is entirely orthogonal to EV's.

Honestly this is a city/urban thing. It's simply not something that on a large scale people who live in the suburbs on in the country will want to do. My vehicle sits practically outside my door at work and in the garage at home. I like the fact that I can go spur of the minute anytime anywhere and don't have to summon the vehicle or do any planning.

I don't even think it's a city/urban thing - I live right in the middle of a city and walk to work but the car is for getting out of the city to walk/bike/ski and I can't see a glorified taxi working in those situations.

Would the parking situation not be amenable at all to metered chargers?

There's a lot of social conventions and what not to work out there, but I doubt any of it would be insurmountable. For overnight charging, a cord that you activate with your phone, with the activation locked to one vehicle, solves many of them.