I disagree with you about the risk of injury is greater in Rugby than American Football. Every sport has some risk. Hell even walking has risk. If you don't want any risk then don't play any sports. Here is where I am different from probably many parents on this board. I think America is ridiculous with how they try to make everything sanitized and risk free. I personally like a little risk, it makes life more exciting. Life is risky, life on this planet always ends in death so no matter how safe you are, your going to die one day anyways. No exceptions.
So I have always let my kids play sports that they chose. My daughter rode horses and got thrown off a few times. My son played soccer, football and lacrosse and didn't get injured at all. The only concussion my son received was when a mentally challenged kid who didn't know his own strength and size hit my son over the head with a stick. So if you want to live life in a sterile vacuum let society continue the way it is. Pretty soon everyone will be vacuum sealed in their livingroom living life via machines. Kinda like a few movies and books that have been written on the topic.