From what I've seen from these large cities, the only reason people think that is that they remain in their small subset, which is large enough for them not to notice the rest.

If you're so inclined, sure, your small rural town is too small to have more than one community, and so there will be a little bit of social diversity. But if you live in a large city and are willing or need to go out of your in-group, the diversity is much larger.

There's a double crux here; Firstly, why would you? By sheer numbers I'm sure there's more school teachers, insurance salespeople and tattoo artists in NY than where I currently live. But while in NYC, why would I have gone out of my way to meet people with these professions specifically when there was plenty of people demanding my time in my circles already? Where I currently live, I just meet them at the gym.

Secondly, this applies to the other side too. While in SF I once people went to a party with a group of people that was very into photography. I never saw them again. Thinking about their perspective, why would they have hung out with me again? We lived far from each other (Oakland vs SF), they had own clique, and I'm not really into photography. The pull was weak both ways. I don't share much with the insurance salesperson I met at the gym, but it's the daily routine that's led to a meeting point, without any side having to "try", chatting, etc.

NYC has diversity by the numbers, but if you talk about an individual's perspective, there's very strong clustering leading to local homogeneity. SF isn't that big, so I'm not even sure if it has more diversity by the numbers compared to where I now live. But the filters leading someone to move to SF are strong, as is thus the pull to cluster with people that are very similar to you based on where you live, shop, workout, etc.