it's the "at least shouldn't" that's what's wrong. if your director sends an email late Friday night and your amalgamation of devices and nervous habits ends up seeing it before 9am Monday, it's going to sit in your brain, messing up your off-work hours. so "send at 9am" is just being courteous these days.

If your own nervous habits lead you to checking work emails outside of work, it sounds like you were preoccupied with thoughts about work regardless of whether or not you received an email. No, the correct solution if you can't ignore a notification is to either block notifications from work apps entirely, or to create a notification schedule that blocks work apps outside of given hours. That schedule should be made during work hours, of course.

If the employee is a nervous person, and that nervousness is impacting their work, and you need them to relax, and they can't, you need to figure out a way to help them relax.