Another example could be needing money for life saving expensive treatment for either yourself or a loved one. That's why I mentioned that "avoiding variance"

Another example could be needing money for life saving expensive treatment for either yourself or a loved one. That's why I mentioned that "avoiding variance" in itself is not a good argument. You avoid it for a reason and that reason is that (u(x - a) + (u + a)) / 2 < u(x) for vast majority of life scenarios.