They need secret shoppers on Airbnb. I have friends that do corporate housing through Airbnb, VRBO, and CHBO. If these fake or misleading Airbnb stories stir up enough problems, local and state governments will severely limit or restrict these listings. There are a lot of people on Bigger Pockets who make a living off of short term corporate rentals.

It sounds like the listing was not up for a substantial period of time. The community did a good job of policing itself.

Wouldn't the address be in Airbnb database when the posting is listed? Why could Airbnb check its location?

Because it’s not profitable to them. It’s absolutely trivial to get purposely-built public housing locations into a database and set up a process to deal with exceptions. But then they would lose their commission.