A quick point-by-point on this. Why are we worried about parents not sending their children to school? If they are good enough parents, we should be happy, surely? And if they are not, then surely your concerns should have been noted before they left school.
Parents who are deregistering to home-ed are enacting their responsibility to provide an education, not exercising a right. If we don't have the ability to educate otherwise than at a school, the state has the responsibility for provision.
A "home schooling register" is a complete non-sequitur, given that we don't use the term 'home schooling' for elective home education here in the UK, indeed @educationgovuk issued a blog stating that 'home schooling' is the term for remote schooling-from-home due to closures.
Therefore, home-schooled children in the UK are actually on a school register. The issues around registering EHE children are detailed here: sometimesitspeaceful.blogspot.com/2016/01/but-wh… Regardless, what a schoolteacher knows about home-education is not relevant, it's like asking a greengrocer...
... about the best cuts of meat. If you want to know about home-ed, talk to home-educators, both parents and children.
On that note, once again, who is the 'we' who don't know where children are? The people who need to know are the parents.
It is frankly offensive to suggest that children who are electively home-educated are more likely to get involved in crime. If you mean children missing education, then say so, and take the available action now to help those young people. A register for EHE is the wrong target.
Please, please make clear distinctions between children not attending because of school closures or infection control protocols, children missing from education and electively home-educated children. If you trust parents to parents, let us get on with it.