I've not read the whole thread, but I suspect it's a lot of bashing of the teaching profession.
I have been a teacher for around a decade in two schools, both very good state schools which are locally very desirable. I have enjoyed teaching immensely.
At first the planning can be all consuming, but in my experience this becomes significantly faster and not at all a problem with experience. Marking will always take time, but this often comes in waves - for example when there are trial exams in multiple year groups all at once there's a lot of marking.
There are certainly administrative tasks, but in my experience, these are not unmanageable for a classroom teacher. If you progress up the ladder, these become much greater.
I think much of what you experience depends on two things, most importantly your colleagues. If your department is well led and the school is well led and SLT places high priority on staff well-being this makes a massive difference.
Student behaviour and culture of the school is second. But generally these things go hand in hand, if a school is well led it tends to have a good culture and good behaviour follows from this. As a new teacher, you will almost certainly experience behavioural problems, even in a great school. If you have a good supportive SLT and department they will help guide you through this, and you learn how to deal with it.
I've never taught in a private school, I think that the stereotype would be that behaviour might be better if it's a good private school, however, there can be more expectation on teacher time. In my current school, it's acceptable for teachers to leave sharp at the end of the day, this may or may not be the case in other schools - in a private school you may be expected to get involved in after-school activities, etc. sometimes the parents at private schools might be more difficult and more demanding, marking load might be greater etc. But these are all stereotypes. As I said I've never taught in a private school.
I think the teaching is a vocation, it's certainly not easy or fantastically paid. Even in a good school, there are challenging students and challenging moments. But if you're going in it to help young people and to pass on your knowledge of maths it can be very rewarding . I find it rewarding to look at their work and see the methods I've taught them and the style I use on the board reflected in what they've written and to know that I've imparted knowledge. It can be a great job.
Overall - get yourself in to observe and see what you think. Choose your school carefully because it makes all the difference.
Good luck