I believe that you are entitled to ask for unpaid leave if you wish to spend your time homeschooling instead of working? Because seriously - you cannot do both at the same time. And the harsh answer to that is that it is not your employers problem what you do with your children. It is, however, your problem, if they dismiss you for failure to attend work or inadequate performance.
I'm sorry, but whilst I do sympathise with the problems that parents are having, those of us managing public services are also really struggling both to maintain effective services and be fair to everyone after nearly a year of this. We went into the first lockdown, as many did, thinking we just have to get through a few weeks of this. A few weeks of "managing" with stuff is one thing. 10+ months is something different.
Even without throwing home schooling / child care into the mix, not a single member of my department would tell you that they work fully as effectively (or as happily) from home. And we have always had home working for some of the time as an option and used it. But full time home working is often not as effective - human interaction is part of what helps people work effectively. If you are then also dividing your attention between work and childcare you are not working effectively. Nor are you doing childcare or schooling effectively.
As public services, we are not allowed to furlough. We are still expected to do our jobs, and not doing so is not an option. And it is not fair on colleagues when some people are not able to do their job effectively because they are doing something else that is not their job. Your colleagues didn't sign up for pulling your weight when you look after your children.
Public service jobs usually do have the advantage of being able to allow people to choose to take unpaid leave. which many employers don't offer. That is what we have to offer - if you are doing homeschooling or childcare (and this includes when working from home) then you cannot also say you are working - you must work flexibly (with your managers agreement), take annual leave or take unpaid leave. In the first lockdown we had a little more flexibility and could give some (a minimum amount of) paid special leave for unforeseen circumstances. But no longer.
I'm sorry, but I am afraid I agree. You cannot simultaneously do your job and provide childcare /schooling. If the experience of your employer is that this has led to poor performance and poor service delivery, then they must take action to ensure that doesn't continue. It's crap, but there is little about this time that isn't.