"But there should be oversight. Parents should have to prove their children are following the curriculum for maths, science and English, and submit proof of work etc.
They should also submit termly reports proving that their children are regularly socialising with peers."
I agreed with you until this point. Firstly, there is no curriculum for home education, so there is nothing to submit proof of work for a "curriculum".
Secondly, there are no "terms" for home education, so that isn't applicable either. Also, writing a report does not "prove" socialisation exactly.
Currently, this is just how the law works at the moment regarding EHE, there is no requirement to follow a curriculum, changing this would be detrimental to many children, specifically with SEN. The NC isn't exactly something that everyone agrees in terms of it's content.
When known to the EHE department, we are usually asked to provide annual reports containing progress. At the moment, any contact more than annually, other than formal, would be unlawful as it would be classed as monitoring which is unlawful in the UK. I'm just saying what it is now, not saying this won't change in the future.
As a former teacher, and now home educator, I find a lot of comments on these threads are written by people with a lack of knowledge of the current government EHE guidelines. They often have a distinct lack of knowledge of home education and how it works.