OP, I've had an idea, having reflected on what you have posted so far about your preferences. You mention that you own a house other than the one you live in. You also say there are lots of independent schools near where you live now. Why not stay where you are, sell your other house, and use the capital to fund independent schools for both your children? You could take financial advice about how to handle the proceeds of sale to maximise their investment potential and be tax efficient.
Hopefully, there is a small, nurturing school close by that you can walk to to drop them off and pick them up until whatever age you deem suitable for them to be independent, so you need never be late and the risk of them talking to any adult other than you and the school staff will be reduced. As you would be a customer rather than a tax payer as in the state system you so dislike and distrust, you would have much more say in how the school managed your children day-to-day and they would get much more individual attention because class sizes are nearer 10 than 30. My DD went to an independent school for A level and was in a class of two for French. The holidays are longer too so you would have more time at home with them.
Being a customer also has its downsides, of course. If your children had behavioral problems, for instance, the school could take the view that they would rather be without both the behaviour and the fee income so your children would be asked to leave. You would also find higher expectations of you and your partner participating in school events with other parents, which given your preference for a solitary, domestic life may not be welcome. Another risk is that your children might make friends with other children who live in neighbouring villages, increasing the risk of play dates and parents offering your children lifts.
Obviously this would all depend on the value of the house, and how much, if any, is mortgaged. There are hybrid options too - state primary out of catchment, independent from 11 or 13.
The key point, though, is that there is no perfect solution for any of us when it comes to choosing a school for our children, and we have less control than we may like to think. My DD went to an out-of-catchment infant school because I didn't like the fact that the school opposite our house at the time was racially uniform (every child white, as we are), unlike the civil service nursery she attended and loved. My neighbour was unhappy about classes being shuffled around at junior level. I told him class composition is the school's to decide, because they know all the children well.
As several have said on here, home schooling presents difficulties too - cost especially at secondary level, risk of isolation and poor socialisation of children, burden on parent, narrow curriculum, need to evidence to the local authority what has been taught, the forthcoming duty on local authorities to maintain a register of home-schooled children (see previous today).