The role’s listed requirements include a “background in urban planning” and a “virulent vehemence for vermin.”

This line is the Key which unlocks the context lock.

"background in urban planning"

It's talking about gang rats.

Rats are snitches. Snitches get stitches. Who gives people stitches?

The Rat Czar.

This is a casting call for someone to step in and keep the NY gangs (urban youth) in line. Gangs deal with the government all the time, in terms of FBI deals.

Consider the rat at the end of The Departed. Those are the established Comms. In big cities, everyone is a rat. The rats go in a circle, chasing one another's tails.

Eventually, all rats get a bullet in the head. That's the "moral" of the movie anyways.

https://youtu.be/V3f9UJTmgd0?t=64

Again, if they are hiring a Rat Czar, they're looking for an enforcer to, effectively, kill gang whistleblowers -- anyone in a criminal organization who is willing to come forward and say "The mayor of New York buys drugs from me" or other some such incriminating stuff. There's so many vectors of crime, it's hard to say it's just one variety. It goes all the way from the bottom of the totem pole to the top, from the standard beat cop up to the DA and beyond.

Which is why you need a heavy hand to keep all the rats in that circle chasing their tails. If the conga line finds a snag, all their gooses are cooked.

Considering what you know about Chicago crime, this line from the article should seal the deal on my decode:

The city is topped by Chicago in Orkin’s rankings, which has held the “rattiest city” title for eight consecutive years now — but New York rodents have a special place in the cultural zeitgeist.

Replace rat with gang informant and you got your hidden message. Considering how on the nose and down-to-business this article is, I'd say it's a pretty desperate call. The $170,000 per year salary also indicates it's going to include "nuclear" levels of information. In other words, information with enough energy to level a city.

Hope that helps.