>And while supply did re-enter the long term market, it still doesn’t capture if it eased rents. We can measure this by comparing rents in the Airbnb-heavy neighborhoods and see if they fell or grew more slowly after the ban compared to neighborhoods with almost no Airbnbs.

No, we can't. People will look at multiple neighborhoods to rent, so the effect or more homes available and lower (than would be otherwise, not necessarily in absolute terms) rent would be spread across neighborhoods, whether they have Airbnbs or not.