> If you got into a DUI, how does the government get to tell random website X that they are not allowed to say that you got into a DUI, especially when the information was public record.

Like every other countries in the world have?

> how does the government get to tell random website X that they are not allowed to say that you got into a DUI

By saying you can't publish the information and suing if people do? It's not perfect but its very easy for governments to not make it show up in google so employers won't know, like it's a right in almost every developed country.

Americans seem unable to imagine anything else than what is.