I was answering to your comment, which seemed to imply that they are never point-to-point, when the system as a whole often are. And for the on-demand thing: If it leaves every 10 minutes or so, that's close enough.
I'm not trying to imply anything, I want to proclaim it boldly! Trains are not point-to-point. They don't go from your door to a destination, and don't generally provide direct routes. They don't depart at your convenience, and in the US, at least, they don't depart every ten minutes. In fact, there is virtually no reliable subway/tram/train service throughout the US except for a relatively small number of cities - more on why below.
Even where trains are available, it's not what I meant by point-to-point.
When I lived in Boston, I took the subways/trams everywhere. However, the system is laid out as spokes, and to go to some destinations I'd have to ride all the way in to city center and then out another spoke. It made what would have been a 10 minute trip by autonomous vehicle into an hour to hour-and-a-half trip (depending on how the trains were running that day) with transfers. I chose to commute to a summer job by bicycle, 20 minutes each way, rather than lose 3-4 hours out of each day by having to ride the spoke system, transfer to buses, and then still hike a kilometer from the nearest bus stop to the job. However, biking wouldn't have been viable in the winter, nor would it be viable now that I'm older and missing the cartilage and half the ligaments from one knee. But part of my argument here is that the public transport solution wouldn't be viable for me now either - hiking a kilometer on ice and snow both ways daily would now be both painful and dangerous for me.
But shure, if you live out on the contryside, it gets harder, especially during the night. However it's funny that you mention airports, as they are often quite well served by public transport. I have myself several times taken the airport express train or bus to catch an early flight.
The fact that one end of the system is tethered to high-volume points of interest doesn't help people who live nowhere near any of the routes that go there.
The US is very large compared to Europe, and most folks here don't have access to transport systems comparable to those in Europe. Building the infrastructure isn't viable for most of the country because we have much lower population density. For better or worse, these are the reasons why the US has a more car-oriented society. I think autonomous vehicles can potentially have a huge impact here via pooled usage, reducing the need for individual ownership while still providing the point-to-point and on-demand benefits that are currently only available with privately owned cars.