TeenWolf,
It seems to me that, essentially, all parents who HE do so due to a mismatch between what the 'mass education system' provides and what they want for their children.
For those who - often reluctantly - HE their SEN children because of lack of suitable support in mainstream or provision of Special School places, it is quite simply a scandal, and a register system combined with some indication of 'reasons fro HE' would be an effective way of revealing its scale and scope and thus allowing everyone to obby for it to be changed.
For those who HE to maintain a particular cultural identity or 'micro societal norm', again registration - not approval - would show the scope and scale of the need for sensitive mainstream education or educational outreach.
For those who HE their children because of bullying or educational problems in the mainstream school they last attended, registration would allow Ofsted to be alerted and inspect these establishments (Many state schools are good. Some are not. Failure to protect, or provide for the educational needs of, a child so that their parent decides to HE them should be a factor that inspectors take into account in triggering an inspection and reaching a judgement).
Then there are those who HE out of conviction - those who find the structures of school too rigid, the expectations not age appropriate - and again, registration can help to start to sift these from the tiny, worrying minority who home educate as cover for neglect.
Registration, done properly and sensitively, should help HE to eradicate the stain of the tiny minority who use it as a cover for abuse or neglect. Done badly, it could further erode the already fragile trust many HE families have in 'authority', and allow the small minority of abusers to hide even more effectively.