I agree that it's probably the staffing ratio as well as setting precedent. If Friday is the slow day, they may have fewer staff on and have to make sure that they have enough teachers to accommodate an extra child. It's definitely not a call that the teacher can make, and she may be thinking that if they let your child come in, then they would have to let other children in the same situation come in as well, etc. The other thing is that daycare centers rely heavily on routine (it's the only way to get through the day) so having a child in the classroom who is not normally there can mess up the other little kids. It's not the end of the world, obviously, but there are many good reasons for trying to keep everyone to their agreed-upon schedule.
That said, I used to ask every now and again to add an extra day for one of my children if their other care provider (my mom) was unavailable on a day that they normally would have been with her and if other children were scheduled to be out that day, the director would allow my children to go but it was a decision that she would have to make. So I don't think it was out of line to ask, but there are valid reasons for them to say no. Also, the "I paid for 2 days I should get 2 days" doesn't really work, unfortunately. The center probably still had to pay their teachers even though the center was closed. It's lousy, for sure, but I pay for daycare whether or not my kids actually go and that's just the way it is.