I also know the law inside and out. You are more or less correct in your statements but they are a bit misleading as they leave out a lot and can lead to abuse if misunderstood, and potential criminal penalties or lawsuits for people with fake service animals. I'll fix the statements.
"Not accommodating is not an option." - You are correct that all businesses (and hosts) MUST accommodate LEGITIMATE SERVICE ANIMALS at all times. A business or host CAN refuse accommodation or service (legally, Airbnb terms is another story as they leave their terms intentionally vague) if there is evidence that the service animal is not legitimate.
The service animal is not considered legitimate, by law IF:
A: The answers to the following questions are not satisfactory to a reasonable person on a jury:
1. Are you the person with the disability for which the animal present is providing a service (or any form of that same question)?
2. What specific tasks is the animal trained to perform for you?
B. The service animal is not housebroken.
C. The service animal is not under the control of the disabled person at all times.
In any of the above situations, the host or business provider, LEGALLY can refuse service or accommodation.
According to Airbnb, guests are not required to disclose that they are bringing a service animal prior to their arrival. Airbnb pretty much goes quiet after that.
There is nothing that I have seen in Airbnb terms prohibiting a host who notices a dog that was not included on the reservation, and asks the above questions.
There is also nothing I have seen in Airbnb terms that prohibits a host from questioning or asking to be removed an animal that does not meet the criteria of a trained service animal.
I will provide an example, legally, which could be ground for criminal fraud charges or civil suit (and, of course, refusal of service/accomodation).
Example:
Guest arrives with Fluffy. Fluffy is barking at every bird and child who walks past and lunging at other guests. Guest is barely able to keep Fluffy from bolting the pressure on the leash is so strong. Host asks if this is a service animal. Guest replies yes it is. Host asks if guest is the person with the disability for which the animal is providing service. Guest replies yes. Host asks, "May I ask, what specific tasks is it trained to perform?" Guest acts dumbfounded. Says "It's medical." "It calms me down."
Host makes a note.
Later, Fluffy is seen in the building lobby, running off leash chasing a child. Guest who owns Fluffy is on the floor above having a cocktail and didn't notice Fluffy ran off.
Child hides. Fluffy poops on the carpet in the lobby.
Guest comes running down, apologizes. Says "bad girl."
Host asks guest to either leave or remove dog from the premises as it's clear it is not a trained service animal meeting the ADA definition.
Guest refuses, claims it's a medical necessity.
Police are called.
Camera footage shows dog was not under control of disabled person at all times, was not housebroken, and couldn't possibly be performing vital tasks for the disabled person.
Guest is charged with service animal fraud and subject to civil penalties.
Host is fine, legally and morally.