Brad Bird could make a great Fantastic Four...
But I respectfully disagree.
See, the thing that makes The Iron Giant as a character work just like Superman is that they tie in enough similarities to the core traits of what Superman is and represents and then applies them to the Iron Giant. A being from another planet, blessed/cursed with extraordinary destructive power, who rather than destroy life, selflessly puts the needs of others before his own, even at the cost of his own life. ... ... Well, it worked until Man of Steel, at least.
But the same thing that makes The Incredibles work is what would make the Fantastic Four work. Sure, the plot and setting and characters are different, but the thing that makes the Incredibles work is that they aren't just a superhero team; they're a FAMILY. They've got dysfunction, distrust, egos, immaturity, conflict, and differences, but they still ultimately love one another and stick by each other because they're a family, through thick and thin. That's not just the heroics... it's the other stuff, the personal stuff. Dealing with middle-age, parenthood, paying the bills, clinging to glory days, balancing time with friends and with family, trust in your spouse, dealing a violation of that trust, etc., etc.
The Incredibles isn't a story about superheroes; it's a story about a family... that also happens to have powers. That's the Fantastic Four. They aren't just heroes, they aren't just scientists. They're brothers and sisters and fathers and daughters and husbands and wives and uncles and mothers... Heck, even Doctor Doom in the comics is the godfather of one of their children. They're a family, first and foremost, and that is the part of them that the Incredibles tapped into so strongly and applied so well.