In my experience, a lot depends on the child's understanding of why they are going to boarding school. The people I know who have struggled most in later life are the ones who either:
a) felt that they were being sent away because their parents didn't like or relate to children and found them a nuisance;
b) were taught that they were being sent to boarding school to become (extremely) successful and they might be a bit unhappy but would be grateful in the future when they were (extremely) successful. They are the ones who have never really been able to shake off the 'success at all costs' mantra - to be, or appear to be, a success is the only thing that matters, and everything else can and must be sacrificed to that.
I'm not sure in either cases that remaining at home and going to a local private school would have particularly changed things for them, because I don't think the message they received from their parents or their school about their own value would have been very different.
Personally, I went to boarding school at 7 because I come from a fairly remote group of islands which don't have a school of their own, so all the children have to board at a school on the mainland.
I am fairly ambivalent about the experience. I understood it as a practical issue and was happy at school, but even coming from a tiny island I could see how cut off the school was from everything and everyone around it, and wilfully so.
Unless we returned to the island, I would not send my child to boarding school, particularly at such a young age. I don't think it is good for a child, or the adult that they will become, that almost their whole experience of life is being so deliberately removed from the rest of society.
For we islanders, it was always a great point of principle to stay engaged, politically and socially, with events off the island. We read and learnt and debated and emphasised. At school we were so not just physically, but psychologically removed from the people and society around us that they were no more real to us than events we read about in history books, an idea of people rather than people themselves - we would see them sometimes if we went to the shops, but they were not us, they never would be us, and we would never be them. For some people, that's fine and they spend their whole lives surrounded by people just like them, but it's not what I want for my child.