There absolutely should be a register of home educated children, with welfare checks and checks to make sure they're reaching an acceptable level of educational achievement (adjusted for SEN).
Parents providing a good quality of home education and welfare won't have any trouble.
The ones who will have trouble are those who are incompetent and happy with illiterate 11 year olds, those who start "home educating" to avoid prosecution for poor school attendance, gypsy / traveller families who remove their children from school early and label it "home education" while providing no education whatsoever, and those who use it as a cover for child abuse.
Just because you do home education properly it doesn't mean everyone else does.