Children fall under several categories. I don't think they've been understood here, forgive me if I'm wrong. I know the media has gone a long way to cast HE families as shady types but there is genuinely already a system in place for identifying us who actually home educate.
Schooled children - sent to school daily.
Home educated children previously registered at a school - reported to LA and are already being monitored. (This makes up the vast majority of HE children, my own family included).
Home educated children never sent to school whose parents are legitimately home educating - either A, reach out to their LA, or B, are active in the community and not invisible by any means anyway. (Seen by Gps, dentists and so on who could report safeguarding on snapshot moments such as the intended annual monitoring would be).
Abused children being wrongfully classed as home educated children (simply as they weren't registered at a school) - kept indoors and abused at home. Not registered anywhere and not seen by anyone at all since birth, if birth was registered. Or not seen in years by any medical professionals they would expect to see, such as a dentist.
Abuse can of course happen in any of these categories and as we know, abuse is often hidden well.
What would further monitoring of HE families who are already under a LA achieve? It wouldn't find the ones lost through the cracks.