I also see some of the same, and some different, anxiety tendencies in my family members. Did I develop social anxiety because of my parents or one event? I don't know the answer to that. I'm not sure anyone can know that. Do we develop coping strategies in life that we may learn from our families/parents? Sure, of course. And maybe those coping strategies we learn aren't always the best coping strategies for different points in our lives - like if we develop anxiety as some form, albeit not entirely helpful, of coping strategy or reaction to a stressful environment.
I also try to realize that on the whole, people are trying to survive the best they know how, to deal with the current circumstances the best they know how at that time. If something happened in my past that affected me "negatively", the intention of that person, if it's my family member was not meant to harm - they were coping maybe the best they knew how.
But to your point - we care for and love our loved ones. We hope they also "get better". But you can only do the best for yourself, that is what you can control and guide - yourself. And in doing the best for yourself, you do the best for those around you. I cannot point out and force my family members to see or address or even discuss their "issues", just as no one could have forced me to do therapy before I was ready to make that choice for myself. But, you can be a person who is trying to do the best for yourself and those around you, and then be available when someone comes to you for help, advice, or simply to be listened to.