This is the sort of question for which we'll soon be adding the ability to ask owners of a specific car about it.
I happen to own a related car, the Ford Taurus X. The cargo space is just adequate for our family, and we have three kids, and probably pack more efficiently than most. When we really need space, we can put everyone in the first two rows and fold the third.
For the amount of space you likely need, the only viable alternative to a large SUV is the one you probably don't want to hear, a minivan, which can carry at least 50 percent more than the Flex behind the third-row seat.
If you must have AWD, then you have one option, the Toyota Sienna. If you don't need AWD, then the Honda Odyssey and the current Kia Sedona are also worth a look. The Kia has a little less passenger and cargo space than the other two, but is the most stylish and looks / feels the least like a minivan. The Honda has the most interior space, but has the most plasticky interior. I've reviewed all three (see links below the photos).
The Honda and Toyota have good reliability records. The first two generations of the Sedona weren't very reliable. We don't have data on the third. In general Kias can be a little glitchy during the first model year of a new design, but are okay if not great after that.
Chrysler will be introducing an all-new minivan this summer that could drive very well, but I'd suggest giving them at least a year to sort out glitches. Their first-year reliability record is pretty bad.