Best to repair at your own cost if you are able to or your insurance premiums will go up for the next few years! It doesn't work like car insurance - if your tree falls into next doors and they have damage they have to claim on their insurance if it is something they can't afford to put right themselves😞 If anything is not attached to your house it probably wont be covered, ie, fences, hedges, gates.
During the 87 storm, we had a lot of damage to an oak tree, probably similar in age to yours. The tree was in the lane beyond our property. After the storm the local council sent round tree surgeons who dealt with all the branches which were damaged. All I can say is, we don't let it worry us too much these days. You could get a tree surgeon to survey the tree and take off any branches showing signs of "old age" and get them removed so they don't break off maybe.