I'm not too surprised. You do want to wait a little bit, because you want the swelling to go down. It's quite possible that there really aren't any appointments available until then.
There isn't much they are going to be able to do to help him other than splint it. When it comes to other bones, they splint the bone right away and then cast it once the swelling goes down. Maybe once the swelling goes down they will give him a different splint. I'm sure they doctor will want to look at it and double check to make sure nothing was missed and the finger is in the correct position to heal
It really is ok to wait a few days. You could always call the doctor's office to see if there are any cancellations. Your son is going to be just fine waiting until Friday.
And Wild Woman, seriously? You don't have to love Obama, but people are not waiting longer for appointments because of Obamacare. It's fall. People are active. This doctor just happens to be busy taking care of other patients. Sheesh!