That's why you have teams, I was on the senior level team. We shared the responsibility; thus, I shouldn't have to check-in. Other senior engineers went on vacation during my tenure there without looking at a single email or picking up the phone, so why should I have to be held to a different standard?
You can't 'exit a system' if you have responsibilities, in a situation wherein that machine could break down in your absence.
If 'reading a few e-mails' is 'stressful' - then the person has issues working in a professional environment. 'Checking in' should not be stressful. It's just that - popping your head in to see what's up.
In reality - there is absolutely no escaping this for those who have responsibility unless there is a lot of planning, especially the further 'up the chain' you get.