I would jut let them know that you have an appointment that you booked on a day that all families said they did not require care. This means you do not need to book the appointment January 4th when all families do not need care, that it was for their best interest.
That said, unless the family is just all around disrespectful I wouldn't really consider this a rude thing to do. My husband would likely have asked the same thing (making me cringe) but he is of the mindset there is no harm in asking...you really would be surprised at how much money you can save by asking such things. Drives me up the wall but my husband continues to do it because he continues to receive money in exchange for asking in such cases. That said, if we had a caregiver for our daughter the caregiver would understand it wasn't an insult as they would know they are respected...which doesn't seem to be the case here.