The minivan's job is to haul people and cargo in as comfortable and efficient a manner as possible, and it fulfills that mission admirably. Forget about three-row SUVs. Minivans can carry more people more comfortably than even large SUVs; and with the extraordinary flexibility of seat placement/folding/removal, minivans are unparalleled at virtually everything you'd need it to do.
Need to carry a bunch of kids and their stuff on a road trip? There's no better vehicle than a minivan. Want to haul as bulky a load of cargo as you could in a pickup, but you'd prefer to keep it dry, clean, and secure? Fold down or remove the seats, and the minivan becomes a cargo hauler par excellence. Want to take your buddies on a week-long backpacking trip? You can fit everybody, their backpacks, the food, AND a few cases of beer.
People rightfully rave about the cargo-carrying flexibility of wagons, hatchbacks, crossovers, SUVs, and even oddballs like the Honda Element and PT Cruiser, but all of those pale in comparison with the humble minivan.
As a group, SUVs' sole advantages over minivans are style and sheer off-road capability--and it's not as if today's popular car-based SUVs (effectively minivans in drag) are fantastic at low-range bouldering.
The point about cargo capacity is a very good one. My parents recently had to clean out the storage space holding what was left of my grandmother's furniture and other large possessions after her death. In the past, they had rented a U-Haul trailer, but that wasn't a whole lot of fun to drive 600 miles over mountain roads with the constant threat of freezing precipitation. So they rented a minivan and took out all of the rear seats. It performed admirably. There was more than enough room to haul what they needed to haul, and it handled as well as a car. And the fact of the matter is that a small-sized sedan just isn't going to cut it for a family with small children. There just isn't enough room to haul the whole family plus passengers, and that's a real pain in the butt when the children are old enough to be in soccer or other activities. The minivan may not be cool, exactly, but if its utility is unmatched, why shouldn't people buy it?