I home educate my dcs and have done so for at least 4 years. It has been the best decision we could have made for them. They both had many difficulties at school due to SEND and lack of support. The schools did nothing about the bullying they experienced.
The LA are aware of us, and I turn in a report each year when they request an update - this year's report was 7 pages long so it had a fair bit of information on it. I always decline meeting with the EHE person as my older dc has an EHCP, so we meet up with the SEN caseworker to do an annual review as needed anyway. We are also in the process of getting an EHCP for my younger dc.
My dcs are autistic and do not cope well with people coming to the house and expecting them to discuss their education or "perform" for them. It's particularly stressful, and I'm not willing to put my dcs through that. We do, however, regularly see multiple professionals (GP, asthma nurse, dentist, paed, PT, eye doctor, etc) as needed, so my dcs are in no way "invisible." These professionals are all well aware that we home educate and are supportive of that decision.
I'm curious how much information or exposure these people that are critical of HE actually have - I don't mean just from one source, but how much research have they done so that they are really well informed? Most of the home educators I know are very careful to make sure their children get a well rounded education as well as a full social life and are taught important life skills. Yes, many are working at various levels at various ages, but that doesn't mean they are not progressing or being educated. Children progress at different rates both at school and in home ed.
The laws in place right now give the LAs ample tools to deal with HE appropriately. I think the government needs to fix the schools that are failing children across the country first.