Being the father of only one child, a 17 year old daughter, I have not had to consider the question about a child playing football.
My daughter is a pole vaulter which is a very dangerous sport. Olympic Champion Jenn Suhr suffered injuries to her ribs, wrist, neck and hands after her pole snapped recently during a training jump.
https://www.youtube.com/watch?v=UbWVoCac_-Y
Penn State pole vaulter Kevin Dare was killed during a meet in 2012 when he became disoriented during a jump.
http://usatoday30.usatoday.com/sports/college/stories/2002-02-23-dare.htm
My daughter (now a senior) first started jumping in 7th grade attempting heights of five feet and the thought of injury was not a great concern. She has now cleared 10 feet twice and is working at becoming inverted during the launch.
She has not been seriously injured vaulting although I would assume despite normally landing in the pit feet first she has probably jerked her head and neck a few times.
Her most serious injuries have been a broken hand playing field hockey, stress fractures from running during training and another broken hand when she went ice skating with one skate and one walking boot.
I do worry at times about her being injured vaulting, but my faith helps give me some peace of mind. My daughter trains year round for vaulting including during field hockey season (she is icing her leg right now between two-a-days for FH). Her passion for vaulting has given her discipline and has helped her to set goals both academically and athletically. Some of her best friends are in her pole vault club and their is a high level of mutual respect among pole vaulters even during competition.
Like golf, vaulting is very technical and can be very frustrating at times. Her junior year was not quite what she hoped for but she and her coach have a game plan for training and I could not be more proud of her
perseverance. As others have mentioned about their experience playing football, the life lessons my daughter has experienced from pole vaulting and field hockey will serve her well in life. She hopes to vault in college but has the mindset that she is giving it her best effort and will have no regrets about the final outcome. As a parent, that is all you can hope for in various facets of life for your children.
As for football, hopefully young players will continue to be taught proper techniques for tackling which will not eliminate injuries, but hopefully reduce the number and severity.