Europe definitely has better public transit. And parking is a problem in an urban area (though that issue goes away with a taxi or self-driving car).
But I think you're glossing over a fair bit with your description:
> When you can pop into a metro that is ~5 minutes from your flat, be on the train in less than 4 minutes, and be within ~5 minutes walk of wherever you want to be
...assuming that where you want to be is within 5 minutes of a metro station on the same line that's near your house and boarding at that particular 4-minute interval. Most trips between two arbitrary points involve at least one line change, which then involves another around of waiting; depending on the nature of the change, it may also involve a walk between stations. That adds up.
I'd like to have a public transit infrastructure that's comparable in latency to getting into a car sitting in your own driveway, going directly to your destination, and parking in the readily available free parking at that destination. I'd love to not have all the wasted space associated with driveways, garages, and other parking infrastructure. But I don't see a path to that even with European-level public transit.
You mentioned Barcelona, so I pulled up Google Maps, and picked a couple of random points roughly 20 minutes apart (by car) within the city, one of them a major landmark / tourist attraction and the other a residential area. Here's the map: https://goo.gl/maps/VCzEJEn3q3k . 24-35 minutes by car, depending on traffic. 67 minutes by public transit, including 20 minutes of walking, one line change, and 40 minutes of transit.
Here's a second example: https://goo.gl/maps/MWGqrSMsFGp . Directions between an event center and a shopping center, both in downtown and near major transit lines. Much better case for public transit: 29 minutes by transit, with no line change. By car: 12-18 minutes depending on traffic.
At least in the longer wait to travel via transit, if you're used to the trip, you get to read or listen to music, and just watch for your stop rather than the constant vigilance of driving. But that doesn't make up for roughly 2x travel time.