Some jurisdictions collect and remit state taxes, but not city or local taxes.
Here is the page that shows all of the taxes that Airbnb remits for all of the different countries, provinces, states, etc. around the world:
https://www.airbnb.com/help/article/2509
The host is responsible for remitting all of the taxes that Airbnb doesn’t remit.
Some hosts don’t know that if they turn on Professional Hosting Tools, they will get a new field in the setup where they can add additional tax info. After adding the tax percentage in the field, the taxes will be calculated for each booking, itemized for the guests, and included in the total before the guests book. I suspect your host doesn’t know about this setting.
As to your point about hosts declining the booking for being “too cheap”, I would say that it isn’t normal practice. I have eaten the difference on the rare occasions when I have forgotten to set prices far into the future. The window for booking is usually a moving target, like 6 months out, 1 year out, etc., so it’s easy to forget to do this as time goes by and new dates open up. However, I always encourage hosts who have come here with this problem to act in good faith and honor the price they have advertised.