Child can learn at own pace. They never get left behind. I know SO MANY PEOPLE who didn't understand some concept in maths aged 6 or 7, and never ever got the hang of it after that it just wouldn't happen to a HE child. There are so many people who come out of school functionally illiterate. They weren't ready to learn in year 1 or 2, and after that it gets more and more difficult to keep up with everything, because so many instructions are given in writing, and so much of the educational product is expected in writing. This wouldn't happen with an HE child - learning can happen perfectly well through conversation and image until child is ready to learn to read, but you can't do that with a room full of 30 kids.
Similarly, HE children never have to wait. I have vivid memories of a teacher taking me aside and giving me a right royal telling off because of my "attitude" problem in a particular subject, when actually what happened was that he'd spend half an hour every lesson going through the homework ALL OF WHICH I HAD GOT RIGHT. Of course I was bored rigid. What was the point in listening to him? I'd understood this stuff, and he knew it because he'd marked the homework. That never happens to a HE child.
HE families don't have to follow the national curriculum. They can follow the interests of the child. If it's all about dinosaurs this month, it can be all about dinosaurs, and that can come into maths and writing and art and science and history and old uncle tom cobbley and all. You can catch the questions of your child when they happen, whereas a physics teacher faced with "why do clouds stay up?" will have to say "er... actually today we are doing ticker tape timers" and the moment is lost.
HE families do not have to cope with bullying. They can truly have a zero tolerance attitude. In adult life, if someone bullies you, you get out of the situation. HE children similarly have freedom of association, a basic human right denied to all school children who, by definition, have to mingle with a given group of people whether they like them all or not.
HE children have true socialisation. Instead of being with a group of people whose birthdays all fall within the same school year (think about it - how many good friends did you have at school outside your year?) they have a much wider cross section of ages and abilities and classes and experiences to be friends with. Just like adults have, in fact (because, let's face it, not all my friends are within a year of me in age, and neither are yours). They are out there in society, mixing with check out ladies and bank clerks and other children and the gas meter man and bus drivers and that old man who walks his dog round the block everymorning at 1043 precisely, at one mile an hour, and everyone else who makes up local society.
HE children make friends with other HE children of all sorts of ages. A 15 year old has so much to offer a 12 year old. A 6 year old has so much to learn from an 8 year old. And that's normal, natural, accepted in the HE community.
HE children do not have to live on an externally imposed timetable (unless their parents mimic the school day). They choose their commitments and they learn to keep them (brownies or football or recorder group or whatever it is). They are well prepared for the independence of adult life.
HE children get to go to all the cool places when they are not jam-packed in the school holidays Not only museums and zoos and parks and swimming pools, but also cheap holidays in June or September - because HE families don't have to follow school terms
HE children go to true experts when they get passionately interested in something. Where a schooled child is pretty much reliant on Mrs Bloggs the Teacher, a HE parent faced with something beyond their ability or knowledge will either get knowledgeable (everyone is learning in HE) or use their networks of friends and family to find someone with real expertise who can help. I know HE children whose parents knew some who knew someone who knew a really famous scientist, who was delighted to spend an afternoon discussing evolution or astrophysics or whatever it was.
HE children get to socialise just as much as they want. In most local areas there are active HE groups, and truly, if you went along to all of the joint learning and social occasions, you'd never spend any time at home at all.
You can stop whenever you want. If your child wants to go to school at 6 or 8 or 13 or 16, that's fine. Quite a lot of HE teens go to 6th form colleges for A levels or other higher qualifications. A LOT do Open University courses in their teens. There's nothing holding them back, you see.