Altering the insurance laws so that each person has to bear the insurance for the risks in the venture that he or she had opted to do would help to reduce this modern blame culture. This would also reduce the insurance burden on organisations (eg councils) who have to deal with claims of accidents which happen in life, and save much litigation. .
Life is, and always has been, a risky business.
I think that you should, if you wish, insure yourself against those risks, not rely upon other people's insurance so that you can sue them. Only use other people's insurance when there is no option but to use their services.
For example, going to school is more or less compulsory, so you have to rely on other people to look after your child. Hence the school should have insurance to cover this. But opting to send your child on a school trip is your own choice and therefore you should bear the risk and insurance.
Branches do fall off trees, conkers do split; there have always been risks in most things that we enjoy doing. So let us take out our own insurance to cover these, if we feel it necessary, don't put the onus on other people. Only be able to sue on other people's insurance when there is no choice but to use their services, eg at work or at school. Accidents happen, so let us live with this fact and insure against it ourselves. We are all human and it is human to err, even if there may be unfortunate consequences for someone else.