It's like when a major league sports team wins a championship. The city goes nuts and top players on the team become local heroes. Like a couple of years ago when the Raptors won the NBA championship, "Kawhi eats here free!" signs were in the windows of almost every restaurant. Now, for saving the city, multiply that x100 for the Avengers.
When Clint and the kids were in the restaurant in the first episode and the waiter said no charge, that totally rang true to me.
I think it's a pretty accurate representation in the show. Clint's profession for most of his life was to be a "ghost." Now he's a celebrity and he doesn't know what to make of it.