There is no scientific evidence that icing promotes healing. -->NONE<-- So, the only reason anyone ices is reduce the pain. If a 12YOA pitcher needs to be iced for the pain, then that pitcher needs a lot more than frozen water--she needs to be rested for a month and needs to find a new pitching coach.
Finally SOMEONE stating the truth about icing. My DD rarely uses ice. I always read softball boards about icing and even had my DD following it for a while but then did my own research on the subject and found that what you say is true.
Rest, proper stretching and massage go a lot further than ice.