"So, I don't think that the choice of subjects is terribly important, once you get past the basics of the "3 Rs".
But parents and teachers who either indoctrinate children with their views, or "teach to the test" and discourage them from asking awkward questions are a bad thing."
I absolutely agree with this. I will consider that I've done a good job if my children grow up still curious and wanting to learn, and if they know how to do it. The rest will simply just follow if those two things are achieved.
The trouble is that some (although not many in the UK, I think) HEors HE in order to ensure their children learn only the things they want them to learn. I think that the vast majority in the UK, though, HE because they want their children to learn what they want to learn, and to be able to choose what they want to learn from a broader arena than school can allow.
HE'd children have far more time to visit museums and libraries, to travel around this country and others if their parents have the funds, to read, and read, and read, to watch tv programmes and use the internet than schooled children. They have more opporunity to stumble upon the things that spark their curiosity than schooled children. They won't have their curiosity dulled by being made to learn things they're not interested in.
And don't we all know how easy it is to learn and remember things we've learnt because we are truly interested in them and truly want to be able do them? And how motivated you are to find out the things you want to know - to find other people who know more about it than you do, to look things up on the internet or in books, to use the library etc. It's the same for children. HE'd children don't just have one teacher, they have far more than schooled children because they learn everything from living - from conversing with shop-keepers, librarians, police officers, other family members, the plumber/electrician/builder, the man in the cycle-repair shop.....
Of course schooled children learn like this too outside of school, but the point is that HE'd children have so much more time to learn like this than schooled children do so, unless you are hoping to HE in order to keep them locked up at home, which really does not happen, then your children just will not have huge gaps in their learning.