"I am most surprised that the influence of a good teacher isn't more highly prized."
I tink the influence of inspirational adults and other children (probably particularly older ones but not necessarily) in a child's life is probably more important than whether any of those inspirational adults happen to be teachers, or teachers of that child.
Of the inspirational adults in my life, one was a teacher (until he started teaching me at school, at which point he became a boring and lazy git), one was a teacher at school, and then the others were my parents, various friends of my parents, various adults I came into contact with because of extra-curricula interests out of school. Teachers don't have a monopoly on being good influences, though if one is lucky enough to encounter a "good" one, then it's a gift horse not to be looked in the mouth!
I agree with AMIS. As usual, it's the exotic family which the article is about. I assure you that we have not been to the rim of mount Vesuvius...
But it's also a no-win for home educators writing about their experiences in public, isn't it? Either they write about the unusual things they do, and the responses are sneers (like in the Times comment thread about how Italian isn't important) or the responses say that they are obviously privileged and HE is not for normal people, or they write about the normal things they do and people say "well, you can do that perfectly well in school" or "my children go to school AND we do all that stuff too". Writing about my family in the national papers? Not my bag, baby.