When I started this post, I didn't realize how this thread would evolve. Thanks everyone for posting your perspectives (especially those who do have legitimate service dogs).
A few days ago, I got an Airbnb inquiry from woman with a service dog. She politely introduced herself, her husband, and her service dog. Told me a little bit about themselves and the service dog including what the dog does for her, the age and sex of the dog, and promising to follow any rules I had about the dog and the furniture.
This is the first time I have had a person with an actual service animal inquire about our cottage and it happened exactly as I expected (and hoped) it would. She was up front about the dog, stated what it did for her, and didn't try to bludgeon me about not being able to refuse her. I pre-approved her inquiry immediately.
The interaction with an actual service dog owner only reinforces my opinion that the woman who prompted me to start the thread was simply a pet owner trying to "game" the system and get her pets past a no pets rule.