If you are currently concerned about the education of a child, then their family should be reported to the LA. If you are currently concerned about their wellbeing, it should be reported to SS.
I'm not against registration - I think it would make sense for the primary, secondary, and in-year transfer to have home education options and for parents to do so at the appropriate times (this would also save the issue of kids getting places they aren't going use and the whole needing to tell multiple people about that). In the most recent consultation on home education, I wrote on this and the needs for better systems to put forth work and public connection for home educated students (as I've been told by my LA that my reports and my children's portfolios pretty much sit in a folder in case someone asks to cover the LA's and our butts, this could be made easier and electronic). We don't have that currently, so the first port of call is the LA if someone has a concern and they can intervene. HE parents aren't immune anymore than any other parent. The way many home educators are registered is because someone (quite often a medical professional) contacts the LA about a child not in school.
I think it is important to remember that there is not one home educating community just as there is not one school educating community or one reason why children are missing education. It shouldn't be assumed that a child registered at a school but not going is home education without any other evidence. I was registered at a school and went missing because my mother kicked me out and I was couch hopping and then later because of depression + lack of parental supervision meant I struggled to get out of bed - it would have been pretty bad for me if people just assumed I was being educated at home rather than missing from education.
The law in the UK states that parents are responsible for ensuring a child receives an education that meets their age, ability, aptitude, and takes into account any special needs whether in school or otherwise. There have been legal cases which have brought requirements such as for a child to be able to be a member of their community (though not necessarily the wider world) though I've yet to see a child's interests required to be included. Making the state required to approve (rather than a legal option to be withdrawn on evidence much like with other parenting options) doesn't really fit with the law and would come across antagonistically to a lot of home education communities as seen by the backlash each time this is suggested in any official capacity. Most home educators in the UK are people who have pulled their kids out rather than ideological, many after long battles with schools and the local authority. All children matter, but we have to work with the systems we have and recognize the communities affected, but try to shoehorn into some ideal without considering them.
There is also the issue - in the age of austerity - of placing further burdens on health care by pushing for annual visits with a pediatrician or GP, it would make more sense for that to be a school nurse which home educated kids can already access and are eligible for the same screenings and things as school educated students. Also, many medical professionals know nothing about home education and the law. I'm registered and on a very good speaking relationship with my local LA and SS partially because of how many medical professionals have tried to report me purely based on me home educating. I literally take pictures of my kids before every appointment because of how many false reports against us we've had in the last ten or so years when, after being told that not being educated in school is not a SS concern, they make things up and SS has to record them. I've had to switch providers because they wouldn't let it go even though being home educated has nothing to do with their teeth or other medical needs. I could go on but there is no good reason I see to either put this burden on GP or pediatricians or to assume they would want to be part of this system.
I've home educated for over ten years. I'm quite structured compared to most others I know and I know plenty of adults who were also home educated. Some have the complaints made in this thread and some don't and either feel their social and education needs were well met or feel their social issues are more connected to autism/medical issues/many of the reasons kids are pulled out, some it was a horrible experience. There are pros and cons much as there is to the local school where only 10% of young people will get the standard pass for English and Maths GCSEs and ever lower amounts being spent on students. I definitely think we need a better system like registration or GCSE access (while I understand the call for HE kids to take SATs - even though other parents can withdraw their kids from them - HE parents have to pay for all testing including any GCSE tests and making any test required while we still have to pay for that is basically saying only people wealthy enough to take those tests can home educate) but I think people need to stop placing an idealized school education alongside home education. That's not the option many of us have, many of us are just making the most for our kids of a bad situation that neither of us can control and the LA apparently approves of because it's enabling it to happen.
My parents spent a lot of my childhood bragging about me and my extracurriculars including far too many hours in dance at too young an age which gave me injuries that I will live with for the rest of my life. Their choice means I will never be without pain or have full movement of my body again. I don't think parents need to be approved to put their kids into extracurriculars, I think we can come up with better than that.