I will grant you that schools doesn't always work as they should. Lots of things can be done to improve them but every human institution is imperfect and there will never be an education system that never ever fails a single child.

We can and should do our best to improve schools and making them safe, inclusive and welcoming places.

This is one option.

The other option is not to have schools to prevent kids from being bullied at school. Kids will just stay at home with their parents. Realistically speaking this is going to be financially impossible for many families under capitalism so a lot of children will be left to fend for themselves and not get an education.

The quality of the education children will receive will vary widely depending on the parents, and assuming capitalism is still in place, children of the bourgeoisie will have much better conditions for learning than those of the proletariat. Also, as kids are taught at home they will never learn to function socially as part of a larger group.

Parents will have full control over their children. Discovering things like child abuse will become a lot harder.

This gets rid of the bullying at school problem but I can't see how you can think it is a better option when you look at all it's consequences.