Azure does have auto-off today but it still doesn't make it easy to change that time per-day. It also doesn't have auto-start. You can also group VMs between resource groups and subscriptions since we flatten all the VMs across cloud providers, regions & resource groups.
I used the email analogy cause I figured that would be more obvious for people to recognize the security risk... it's hard to explain things like drive-by downloading for non-techy people.