Can you remember internal school communications before email?
I can, it was only 7 years ago in my career.
If I wanted to give a message to a member of staff, I'd write a note and place it in their pigeonhole. If it was important, I'd staple the note closed and send a trusted pupil to hand-deliver the message. If the message was to multiple members of staff, I'd photocopy the message and pigeonhole each one. Messages to the whole staff would entail lots of photocopying or the missive being placed on the staff notice board. As a
I like your idea of removing the all staff option and having a member of admin responsible for all such missives.
You mentioned the number of staff who can’t access their emails during the working day as they teach full timetables, so who are the people who quickly access emails and locate missing children? ( “quickly sending an all staff email, means that such a child is rapidly found and potential safeguarding concerns can be dealt with.” ) Perhaps you just need a mailing list of such staff
I don’t know how good your email system is but at my school staff chose the all staff option because there were virtually no others set up centrally for us and many staff did not know how to create a mailing list. Plus we could only use the light version of Outlook at home (where we had time and space to do our admin) and this didn’t have the full functionality for creating lists.
Now if you use google apps then they have no excuse.