i'm not "unaccountable" to the LA regarding my children. My children regularly see medical professionals as appropriate. They socialise regularly with children both in their age groups and in wider groups. I turn in a report yearly to the LA detailing what we've studied and what resources and activities we utilise, and they agree that my children are receiving a suitable education and have stated in writing that they are very happy with our situation. If they were not, then they would be looking into it further, involving SS, and so on. These steps are what the law has in place.
I teach the children what I am comfortable teaching, and outsource what is needed.
I didn't enter into home education lightly. As my ds2's school utterly gave up on him and were letting him steadily fall further and further behind in year 1, claiming that they didn't have the time or resources to put someone with him to bring him up to speed with the rest of the class or even differentiate his homework for that matter, I was not about to let my child's education fall by the wayside. My ds1 was in a specialised school, and in order to withdraw him, we had to turn an educational philosophy as well as an educational plan into the LA to show how we intended to meet his needs at home. The LA panel agreed that we were capable of doing so, and approved the move to home education. Children with an EHCP are also still required to have an annual review each year (hence "annual" review) to go over their needs and progress with their SEN caseworker. So my children are about as far from "invisible" as humanly possible.
The "Mrs Spry" comment was utterly uncalled for and insulting as well. You know absolutely nothing about our situation, and just because I state that "you" or "we" as in the general public need not know anything about my child's education doesn't mean that they are "invisible" to the LA or that I make any attempt to stay "under the radar."
We've recently moved, and one of the first things we did was contact the old LA to advise them while contacting the new LA to sort things locally, as well as registering immediately with a new GP and dentist and making appointments for continuity of care. Hardly the picture of someone that is hiding their children from the authorities.
Most parents that home educate are normal people that are simply doing what they feel is best for their child, and most of the home educators that I have met are doing a very good job of educating (and socialising) their child. The very few I know that aren't have already had run-ins with SS and/or the LA and are known to them.
Testing of the parents is ludicrous. Only on MN, where people go on about making people take a test prior to having children, to home educating, hell even to vote! And then bemoan the nanny state! 
Testing of the children would only put them at a disadvantage and destroy the good that home education provides, especially for children with SNs. Many children with LDs and SNs cannot cope with the pressures of school, the forced socialisation, the teaching methods, the testing. Home education means their parents can tailor their education to their needs, help them socialise in a safer and less stressful environment (and without them being bullied as often happens in schools), explore what teaching methods work best for them as individuals, and avoid unnecessary testing (such as SATS). Both my children are happier, healthier (both mentally and physically as they are also now able to participate in various clubs which they weren't allowed to participate in at school because the school couldn't provide any support so that they could attend and join in) and their education is actually progressing better now that they are home educating.
I don't think they need more checks, I think they simply need to utilise the checks they have in place now and follow up more as needed. When they've eradicated abuse by having it all caught in school 100%, then I'll consider whether or not school is the ultimate safeguard.