There are lots of different reasons why children board. It is not as straightforward as you think. When I lived overseas I was friends with a woman who had moved countries 5 times due to her DH's work. Her DD was very upset at moving to our then country, and after about 18 months loved it and made new friends and loved the school. Another move then came up and her DD13 said NO, I am not moving, I want to board. My friend was devastated.
Another friend has a really talented son and he was offered a massive bursary to attend a boarding school specialising in this. I am talking about a place that would make him a top sportsperson. He went and again my friend was heartbroken.
In contrast, at my DC's private school there were many boys (they never send their DD's) who sent their DC to board at age 7 to feeder schools for the top public schools later on. I hear from others that they hate it, and have massive anxiety issues, but their parents still send them. I have to respect one parent who is probably the poshest out the lot who went against her own family and flatly refused to send her DC to board, whilst everyone else around her did.
Quite a few of my friends DH's were boarders and according to them they have strained relationships with their parents and they are quite messed up.
One thing that totally shocked me was the amount of parents sending their DC to feeder schools to get into Eton. These DC were really nothing special, never picked for anything, nor showed any signs of academic brilliance. They were just average, and sometimes not even that. They were desperate to get them into these schools.
These are the DC who end up running our country. As a result I will never ever vote for someone who has been to Eton.