Of course it's the responsibility of the unit commander. But it doesn't mean that the president shouldn't pay his respects to the family with a phone call.
When my brother was wounded in Afghanistan, I was contacted by the Medical Center Hospital Commander personally.
The Commander came onto the Floor I was working on and told me personally that my brother had been hurt and that if I needed to go and see him to let him know and he would make sure that my leave was expedited.
That was a very kind thing for the commander to do and I have not forgotten that.