@tomatoandherbs perhaps I make assumptions, based on having been through public school system, and hating it.
If there is a mental health wobble, it's good to be within two hours in my mind. When you send them to boarding schools, the parents will not be the number one influence, and this can be damaging.
I mentioned rosy, as private education doesn't come with any guarantees. I'm sure you all know this, but some people seem to not consider alternatives, even if it is putting huge financial pressure on themselves. If you can easily afford it, or have 100% bursary, all well and good.
I recently heard a solicitor friend of my parents complain about not being able to retire, as he's still financially supporting his DC, who went to private schools from 4-18. His DC do work, but they have middling salaries, and it doesn't stretch to shooting weekends, skiing holidays, and keeping up with their old school friends.
Similarly another of my parents friend's is paying the private school fees for the grandchildren. His daughter will not consider anything less, but she is separated from her DH (who we all believed to be rich). His supposed fortune was a house of cards, he's now a bankrupt, and she doesn't work. This friend of my parents is rich, but 60k a year school fees has meant they have had to sell some property, and caused them stress.
The above is all anecdotal, and not relevant to what OP originally asked. OP, if I was in your shoes, I'd consider one near home. I wouldn't want my 13 year old the other side of the country.
And yes all DC are precious. Even mine!