At our school, all teachers have separate email addresses, but they aren't given out to parents - not are they expected to be.
What happens is that all emails come in to the office, then the secretary forwards them to whoever they are for.
I only have time to check my email at 4pm each day.
Before school I have a meeting (mon), playtime duty or prep to do for the assemblies etc (I'm the lucky one who gets to play the piano daily in the hall).
Playtime is spent quickly sorting out things from the previous lesson, or getting stuff ready for the next - if I'm not on duty.
Lunchtime, we take the chilren into the hall for their dinners and then grab a quick break for our own lunch. Then it's back to getting stuff ready for the afternoon or marking work from the morning. One day a week I have a recorder club for 25 mins.
After school I have a gate duty, a staff meeting and a planning meeting.
We can't access email at home. And we can only access it at school if we can use a computer that's logged in as me. When I'm on PPA, I'm not in my room at my computer as someone else is teaching my class in there.
There are many, many day that I do indeed get home to realise I've not managed a bathroom break that day - never mind time to sit and read my emails.
Along with emails comes the expectation that we have received the message and have time to deal with it.