I can see this from all sides - I have HEd, I currently send both my children to primary school, and I teach children from a (largely illiterate) community which removes most of their children from the school system at the age of 13 to 'home educate' them (in fact, to use the girls as childcare for younger siblings - the boys occasionally get to stay on for a few more years).
From my professional teaching experience, I would therefore say that not all HE provides good education - and in fact can be used to ensure that children of some communities do not get sufficient education to be equipped to leave that closed community even if they wish to.
From my personal HE experience, I would say that HE can be fabulous, as it can provide an education absolutely tailored to the child receiving it and can balance education and nurture in exactly the proportions that a particular child needs.
I am now, as I say, the parent of two schooled children. In choosing to send my children to school, I weighed up their 'short term' needs in their primary years - which I reckoned HE with a good dose of 'after school' type activities could easily provide - and their 'long term' needs to prepare them for their 11-18 and then university education. For me, and for my children, I felt that secondary schooling with its specialist teachers for specialist subjects (my DS shows an aptitude for Maths - which I could meet long term - along with an aptitude for languages, politics and history which I felt less equipped to help him with up to GCSE / A-level) would fit better than attempting to deliver the same type of education via HE. My children are academic types, who were likely to need that relatively 'conventional' type of secondary education. I therefore chose to send my children into primary school as a bridge into the secondary education I felt would best fit their needs and aptitudes.
There are definitely good arguments on both sides - and the best fit will take account of parents, the child, the child's age and stage of education, the HE and other 'out of school' group opportunities available and the particular school which forms the alternative.