I will start this off by saying I am a teacher, so not only am I aware of workload, but, being secondary-based, I also have far more parents to deal with than primary-based colleagues on a day-to-day basis. Our staff email addresses are visible on the website, so all of us can be contacted directly, and it is honestly my preferred type of contact, because I can do this at any time, without having prolonged telephone calls or meetings to deal with. I get a few emails from parents per week, often clarifying sanctions, discussing homework issues or alerting me to family emergencies which could adversely affect their child. I reply as and when I can, within 48 hours.
Yes, I have had the odd angry parent emailing at stupid o'clock in the evening and one or two who may have been drunk - they get forwarded to the senior leadership team to deal with.
One of my children goes to a school where teachers' email addresses are not publicly shared, but it doesn't take a genius to figure out the correct address. They're in Y5 now, and I have had to email teachers a handful of times in all the time at school, only 1-3 times per year. Every single one of their teachers has been lovely and replied, in work time, to my messages and dealt with the requests or questions.
I cannot phone the children's school as I am teaching all day and even if I were free, the signal in my school is so bad I cannot make a productive call during work times, and obviously the office is shut before I get to work and when I get home. For the same reason, I cannot have a quick chat on drop-off or collection, so email is the only way of contacting said teachers.
I have had to email the class teacher yesterday after my child had some unfinished computing work that we cannot work on from home (it's not on the shared drive or in any app - we looked), but that needed completing and my child was keen on finishing the work, ready for a presentation they need to hold next week, and in their own words, they don't like leaving work unfinished. I have had quite a shirty reply back telling me to go through the office for future emails, and that my child will just have to do the work again at home if they haven't managed to do it in school time.
I am quite taken aback to be honest. This teacher knows I cannot contact them in office hours, the office have a 72-hour working day response time policy for anything deemed non-urgent (which unfinished work likely counts as), by which time my child will have embarrassed themselves in front of the class if the work is unfinished, and doing the work again will take unnecessary extra time, given the effort they'd already made in class. In their 5 years at the school, this is the first time a teacher has been anything but friendly about direct contact.
In this day and age, should teachers be uncontactable directly, with an office front to screen and essentially decide on what is, and what isn't allowed past?