If I was fairly sure I'd be visiting N America then I'd definitely consider United.
The OP is traveling in business class, so the better programs will be LH and BD. And if they achieve *G, it is also better to be with a non US carrier for lounge access in the US.
Maybe the parents can join BD and the children join LH. Jetfriends (the children's program of LH) Miles do not expire. If you can get a LH credit card, then your miles do not expire, either.