I think op, if you know you don't really want this for your daughter, you shouldn't go to the open day. You sound as if you are under a lot of pressure, and fighting for every inch, but IMO unless you are keen you should draw a line in the sand.
This is not a massive decision unless you want it to be. Most families don't agonize over this and you don't need to either. You are her mother, and you will make decisions as you see fit, the end. Thank pil for their generous offer but put an end to it now, don't let it be a 'thing' any longer, giving them hope, you stress, and your DD confusion.
I'm not saying there are not particular circumstances where children thrive in boarding school and it works for the family, there clearly are - unstable jobs of the parents, prodigious talent in music/sport etc - but that isn't what's happening here.
For what it's worth, dh boarded from 7 with no real dramas, had a great time from 13 onwards, has turned out 'normal'
, and has a good job... but he would never ever complate the same for our dc. And nor would I, it would be over my dead body and maybe not even then that a child of mine didn't live with me.
All the best op.