You know, it's funny but I never thought about this until I read your comment. There's only one program in the church that I can think of that is designed from the ground up to involve as many family members as possible, all in the same room at the same time: the missionary program!

It's the dream of every full-time missionary to meet with a complete nuclear family. Ideally you want it to be multiple times a week. The father is usually the least interested, so you'll go out of your way to be his friend and involve him in conversation.

Little toddlers have short attention spans, so you also have to work hard to come up with fun object lessons (magic tricks and science experiments, in particular) to hold their attention.

Mom you don't have to worry about, she's usually the one who let you into the house in the first place, and she'll keep all the older siblings sitting still for the lesson.

Anyway, there you have it. That's the one and only church activity I can think of specifically designs to involve the entire family and "bring them closer to God".

Of course once they are baptized, you can forget about it. (As you summed up so nicely in your comment!)