No, that's a made up, long after cities, retroactive explanation.
Cities were made so people can co-live and co-operate in larger scales.
In ancient cities people (the people who made the first cities) weren't isolated at all.
In fact, in a city like Athens all life was public, in the sense that private matters didn't count culturally as something worthy of pursuing - it was all about contributing to public life, to be seen, talk, have fun, discuss, vote, participate, in the agora (public market), the pnyx (voting place) and other such places.
The same goes not just for ancient cities, but for most cities well up into the 20th century (just not ones dominated by car designed roads, large non-walkable distances, and/or scyscrapers).
Even in a place like New York, you very well knew your neighbors, and there was a community around the area, street, etc. Kids played together (80's suburbs style), grown ups collaborated, had fun, and gossiped with one another, etc (for example check most accounts of the ethnic neighborhoods, Jewish, Italian, Irish, etc. in 20th century New York).
Much more so in older cities, in Europe, Asia, Latin American, and so on, where most of the living neihborhoods where built as smaller communities, with small low-rise houses, central gathering places, and so on, and people knew each other.