I grew up in a small town and my Dad knew everybody in the county and the next one, whether they were in town or in the countryside. And likely a couple of generations back. Several families moved to California, and the social network reached there.

A cousin of mine had a brother that fled home. He eventually tracked him down using old shoe leather techniques.

My experience and that of my family does not match this theory of "loss of anonymity."

Now, modern suburbs are more likely to be anonymous, and perhaps that is where this theory comes from, but it does not come from small towns.