Even if the school could contact me straight away at work (which they can, they have my work number, dd1 also knows it, plus my mobile works at work) I can't leave the shop unless my boss is in, which is rare because I am not authorised to lock up, nor do I have the keys to be able to. So I'd rather not know that my child is injured and be stuck not being able to help. I'm not going to walk out of employment for a child with tummy ache. If I'm out they can reach me but are often advised to ring gran because she drives and can there quicker than me if I am out.
I trust my Gran to deal with things competently and she is happy to be responsible for being the main emergency contact for the school and is aware that this might involve taking either child to A and E. Anything life threatening/serious and I'd walk out of my job, obviously. My Gran has been the emergency contact long before I had issues with my phone and will be long after they solved because she does not work and she drives, making her the more sensible choice, I ring her to take us to hospitals ayway if there are accidents.
I honestly do not get the big fuss about not being at the end of a phone 24/7. What on earth do you people do if you go camping in a valley/signal black out area? 
I like time for myself and quiet time and not being at the beck and call of others constantly. 9 times out 10 when people do ring me it is to ask me to do something for them, as in yesterday.
DH left it too late to buy his mum a birthday present and then panicked knowing he only had an hour to get hold of me and get me to go before I had to go to work. He managed it, eventually via FB. No big deal. If he'd tried that in the first instance, as I constantly remind him to he'd have gotten hold of me quicker and there'd have been no panic.
I can't for life of me understand why people feel they need to be permantly reachable, bar terminally ill or disabled relatives, of which I have none, thankfully.