I've had that experience only once in my life, when I lived literally in the center of town in Palo Alto, and ran into coworkers very regularly on the street, because our company gave us a subsidy of hundreds of dollars every month to live within a 10-block radius of the office. And these were misanthropic pale boys who mostly had no hobbies and avoided going outside and meeting other people if they could help it.
I've lived in 200k-sized cities and I've lived in college towns and I've lived in quaint, "walkable" suburbs. And nowhere have I had the experience of serendipity as a result of density, because in all those other places I would get in a car and drive to a specific destination point and meet no one either there or en route. And even if there were some amenities I could "easily" walk to within "only" a mile or so, no one else I knew walked there with any regularity, so I'd never see any friends around town.
Objections that go "but I'm happy with my life in XYZ location, how can you say that" are missing the point. If you haven't experienced the difference, it is as stark as night and day.