In the smartphone world, I'm not going to have a mountain of unread emails because I will have at least looked at the subject and first line or two of each of those messages within a few hours of them being sent. How do I avoid having to keep checking? Telling my boss to call me because I am just going to turn off push notifications isn't going to work. That's part of what gets solved here with this proposition. Upper management has cut the cord for me. My hypothetical boss can't say "I know that corporate says we have a great work-life balance, but I'm gonna need you to always check your email". He has to make the call that something is important enough to interrupt my vacation for.
I liken this to Nielsen removing the "reply-all" function from their email in order to curb overuse of CC and rampant extra email that gets in the way of actual worker productivity. Management could preach all day long that you shouldn't overuse it and the employees will just keep doing it until they physically cut off the button. The workaround is easy (just manually add the CC list), but it requires just enough work and forces someone to apply just enough thought that they will now limit distribution to people who really need the email. Same thing goes here: if they really want me, there is contact info in the auto-reply. They can call me, text me, or forward it to my personal address that I might be checking--but its an extra step that makes them think "is this really the best way to get what I need? Or should I try contacting someone else?".