These things never start with your insurance company doing anything wrong. But to your analogy, if you were at a friend's house and someone broke in and injured you, yes, your friend's homeowner's policy is on the hook. If someone walks on your lawn and gets injured, your homeowner policy is on the hook, even if you weren't home
. Nor does intent matter. Whether by accident or intent, if you caused it, you can be found liable and even, in the case of your lawn, if you didn't cause it.
So, there is a legal question here. The guy is liable, does his general liability policy (typically part of your homeowner's insurance) pay or, because it was car related, does the car insurance liability pay. That was decided by the arbiter. It went to the arbiter because GEICO wanted it to go to the arbiter. GEICO lost. Then, despite GEICO agreeing to binding arbitration, GEICO tried to cheat their policyholder (not the victim). The policyholder was the one on the hook for $5.3m as the person liable. This was not a jury trial. The judge looked at the facts, said "this was binding arbitration and there is nothing here that overturns that" and told GEICO to pack sand. As someone who has insurance policies to protect my financial wellbeing, I applaud the legal system making insurance companies honor their policies.
I do take issue with your opinion that someone should have told the woman to let it go. Why? She is going to suffer lifelong medical and social problems including taking meds for life and a significantly increased likelihood of cancer due to the actions of the man found liable. Her ability to date, find a lifetime partner, get married, have kids, etc. have all really been impacted. She has suffered great harm. I don't have the numbers the arbiter used, but arbiters do not generate damages capriciously. If the arbiter came up with $5.3m, that's likely a good guess as to the lifetime damages expected to accrue to the victim. (Actually, arbiters have a reputation for being conservative.) It sounds like you would have given her enough for a single doctor's appointment.