I do know that within the primary and prep schools in an area where a lot go private it is hard as a parent to keep it low key and relaxed. All the tutoring and parental pressure even if they keep it all secret etc. winds up the children with all sorts of unhealthy competitive, and frankly ignorant, attitudes manifesting themselves.
However dismiss this as the schools missing out on a fantastic girl, that they obviously weren't right for her if they couldn't see how amazing she was, lots of hugs, a few treats to make her feel extra special, and don't even mention the two you are waiting on (because that would imply they matter) And arm her for the stupid comments she will get at school. Any girl who needs to show off her result or put others down can't feel very good about herself can she? Whereas your DD knows she is amazing whatever others might say?
And the others are right, this is a chance for her to develop resilience. To gain the skills to pick herself up, and learn from the experience and move on. Plenty of clever children go much further before they encounter setbacks and then they tend to find it much harder to deal with.
Also do not beat yourself up about this either, too many parents think it is all about tutoring. It really isn't, they are looking for ability and potential and the sort of personality they know makes the most of being at that school. Your DD just didn't provide that evidence they needed on that day, or they may genuinely not be the right school for her.
If the Head gave you sound advice I very much doubt the other two are going to come back as nos in any case.