With boarding the decision, surely, has to be a combination of child and school. DD knows a surprising number of "failed" boarders. Girls who did not settle and had to be moved, or boys who are now frantically applying to London sixth forms having left to board at 13.
I don't think there is a hard and fast rule about which schools are happier and which not. Worse perhaps to be in a school where everyone else seems happy.
I recognise the profile of nice, polite, hard-working but sheltered girls, though the ones I know better were at St Mary's Ascot. In most cases I parents have chosen boarding as it provides a gentler environment than West London day schools. We also know one or two who have started boarding aged 13, possibly to take them out of a party culture. (That said my non-boarding DC seem to have avoided parties pretty much altogether, whereas the WA girl DD knows seems to have a pretty frantic social life when at home.)
With overseas students much, presumably, will depend on why they are there. If they are looking for top notch grades and entry to a prestigious University, either in the UK or US, and their parents have sacrificed to give them the opportunity, it will not be surprising if they give a lower priority to wider school life. Especially if the school has selected the strongest academic candidates, those most likely to win the glittering prizes, and not paid much attention to participation in Extracurricular. I might look at the ethnic mix in sports teams and the school play, compared with the mix for Oxbridge entry to help reassure myself that there are not two distinct groups, and then ask how it is managed. (This is a genuine problem, however the worse affected can be sociable girls from a dominant overseas minority who want to mix across the year group but who find themselves pigeon holed from the start.)