If i came to understand that my son, for instance, was not really my son at all, but was really some previous person reincarnated into my son's body, I don't see that as being more 'meaningful', quite the opposite, I'd find the idea very unsettling. The fact that I know my son derives solely from me and my partner makes sense, it is the basis for parent child bonding. if I thought my son was in fact derived from some random other stranger unrelated to any of us parachuted in out of nowhere I can't imagine what that would do to family relationships.
LOL!! You just don't get it do you! What do you mean ...'not really my son at all'!
He will still be your son because his body, mind, name and upbringing is given by you. Even you were in another body earlier but you don't think of yourself as someone else, do you?!
We don't become someone else because we were living in another body before. There are many such bodies and lifetimes we all have been through but we still remain 'us' in this birth.
We Hindus accept rebirth as a normal thing but we don't get confused by who we or our family members were earlier.....though we do sometimes think about it if we come across some special experience with some person or feel some special relationship of love or hate that is not normal under the circumstances. We don't go into a tizzy about it though. Just a casual thought about who we could have been earlier and why we feel the way we do.
Anyway....there are many westerners who believe in reincarnation. Ask some of them how they feel about their children or partners or parents. No big deal really.