You seem to be misinterpreting the study. The $9-10 a month number is for the median zipcode in the US, which is unlikely to be an urban area. And it represents the difference for each 1% increase in Airbnb listings. So a median zipcode with 10% more Airbnb listings would see a ~$100 difference in rent from a similar median zipcode.
Assuming that 'a fifth of the overall increase' is correct, that means that a place like SF can attribute at least a few hundred dollars a month in rent increases to AirBnB, as the median rent has increased by almost ~$2k since AirBNB came into existence.