The rats do indeed lick the sidewalk! Mostly they make use of the trash itself, but will happily lick grease from the pavement if that's what's available. The rats in the cemetery feed on food that people drop (a lot of people eat lunch nearby) and from bags of trash on the sidewalk near the church.

The biggest problem is that trash in New York is mostly placed directly on the sidewalk in bags, not in cans or dumpsters, so it is readily available to the rats.