Actually, overall I love the idea! I think there is tremendous value in this service.

I would just want one additional feature: allowing people to opt-out. A lot of appointments are recurring, take hair-cut for an example. This will work effectively for the first time, however some customers (who actually remember appointment) may started fearing that they would get a useless call even though they perfectly remember where and when they need to go.

So, a simple Press 7 if you don't wish to have automated reminders next time will be a great addition.