A family as the basis for an adventuring party could be very interesting, if the players are up for it. A party of PCs is often essentially a group of strangers when they begin, whether the players are or not.
As noted with the news of a Lego Star Wars series...
Would you find it too restrictive? Everyone has to be related or at least connected to the family.
With command structures often getting loose in RPG parties, would you be willing to play another PC’s child? I suspect a family of PCs would have a loose authority as well. I can imagine The Addams Family as a party, or example, not just because they go on adventures and a variety of supernatural powers would be available but because the kids can run amok and the parents find it amusing up to and including attempted fratricide.
For the record I’ve done this exactly once, as a player in a one-shot Vampire: The Masquerade game at the Nationals where the pregen characters were (a) a Sabbat pack and (b) The Simpsons. Due to that structure it wasn’t a big factor, but it certainly informed our roleplaying.
An extended family makes this easier - “you’re all grandchildren of the King” is easier than “you’re all his children” - cousins aren’t generally as involved in your lives as siblings. The Amber RPG plays with a very extended family, spread enough to be more like a nation.
Borrowing a campaign hook from Matthew Knighton of Distant Stars fame, a weird family like the Addamses also allows more variety. And aren’t all families a bit weird?
An estranged family might help too, so everyone has less of a shared background.