The main answer is probably that they don't need to. They have personal assistants and admins who deal with their appointments, their email, etc. And, believe it or not, some of them really do unplug when they go on vacation.

In a former life when internal company email was becoming a thing, there were execs at my computer company that made one of these minicomputer-based office productivity systems who had their admins print out their email, they'd write responses by hand, and have the admin send the response.