How dare the public use the street they paid for...
In the good old days, the streets were paid for by the public and you were right.
But, for many decades most communities have required developers to put in public streets as a condition of developing residential areas. The streets are baked into the price of new houses, but they really have been paid for by the surrounding homeowners.
Legally of course, the streets are public. But, the ethics of the situation are different. If I paid a high price for a nice house on a quiet street, and paid for that street also, why should i suffer heavy traffic because the county has skimped on maintenance of the designated arterial streets?