Long answer: I boarded in the states from 11 (my choice) and it was absolutely the best option for me. My DC will board in the UK or stick with day schools depending on what seems to fit each.
I'll agree with others that I was far more involved in a much wider variety of sports & co-curriculars than I would have been at home because there were a wider variety of activities offered at no extra cost, an expectation everyone stayed busy and involved, and transportation time and costs were eliminated.
Socially, I found boarding beneficial as a kid who was often sort of on the fringes of groups growing up. Being together so much made for close relationships and made cliques less of a thing. It fostered independence for a lot of the little daily things (like keeping track of where I was supposed to be when with which of my belongings) but also a lot of interdependence and communication with both peers and adults. I remember going off to university and being fairly shocked by other first years not being used to seeking out help from professors or other "adults" themselves, not proactively addressing potential problems or obtaining info to plan appropriately, and often seeming oblivious to roommates' comfort/needs or how to resolve conflict with them.
Academically, I had more structure with less struggle/fuss than I would have at home. Having set, mandatory hours for supervised study (with the level of supervision progressively decreasing by year as long as marks were good) was just the norm, not something that felt like I was being micromanaged by parents or punished. Bedtimes were relatively early (again, it was the norm for everyone around me) so I got better sleep than a lot of teens. Help from teachers and tutors was easy to access so private tutoring was nonexistent. We had Saturday classes at my school and long lunch periods so each day had less lesson time spread out more, which worked better for me than the standard day school schedule back home.
I know your question was primarily about value, but I don't know how to put a number on most of it. That's like so many decisions we make for our kids, I suppose - right now my youngest desperately wants to switch art classes to one that's about 1.25x the price and I have no idea how to put a number on the value added to know if the cost makes sense.