You won’t feel safer in an Awd car that’s much older then a comparable lower mileage two wheel drive car. $10k won’t get you everything you want. Just go for basic transportation. You don’t have enough to work with. Knowing you now have only $10 k to work with, puts you in the basic transportation mode unless you are willing to buy very high mileage. Where is the security in that? Awd gives you absolutely no advantage when you consider how old a car you will have to get especially in the south with no snow. Rain soaked roads are easily handled by good rain rated tires. Best car for you with two kids and only $10 k is a used Accord or Camry. You may not like it, but that’s where your best buy is .
As mentioned above, I would steer away from the AWD Sienna b/c of the run flat tires and their cost and early wear. This is a known problem and if you google it you will see people have gone to extreme measures trying to fit a spare in the car. I had the option of buying a clean used one at a fair price but didn’t due to this.
Another word, don’t get too hung up on the minivan image. We had a caravan when the kids were growing and was the most comfortable thing ever for hauling people and kids. Now we have a CX-9 Mazda that handles great and it is fun, but the 3rd row is only good for someone who is 4’ tall and would need to sit there only for 20 minutes tops. If you are wondering, this purchase was somewhat by design b/c we consciously decided not to haul all the family with us every where we go 
Ford reliability has been taking a beating from CR in recent years. Problems with Sync and its related systems are especially common, enough so Ford is tossing it. They’ve also had problems with their twin-clutch transmission programming and some teething pains with the turbo engines. I’m already in bed now, so not too inclined to go see what CR has to say about the Flex, but I think it was mixed, with reliability being the problem. Of course, used values reflect the vehicle’s reputation so you might be able to afford a Flex that’s newer or has fewer miles on it. As a family vehicle it’s very attractive.
With kids as big as yours you do need a full-size third row. The Mazda5 won’t do. The Highlander has grown, with the new model having a very spacious third row. On older ones it’s not great (if it even has one. The Honda Pilot always looks like a toy SUV to me, but it’s apparently a good family hauler and the Mazda CX9 is all that and quite luxurious, too. I suspect it can be had cheaper than a Honda or Toyota, too. Mazda has been making really good vehicles, but don’t hold their values like Toyota and Honda. The Chevy Traverse also seems pretty decent and quite roomy. The really big crossovers/SUVs are probably overkill and most get really lousy gas mileage.
If I remember I’ll see what CR has to say about the Enclave. I agree that it looks great, inside and out. It’s closely related to the other mid-large GM crossovers like the Traverse, and they don’t have the reputation of being major dogs. The Hyundai Veracruz and Subaru Tribeca are two obscure vehicles you might keep an eye out for. When the Tribeca came out it had a weird front end, quickly redesigned, but Subaru has never sold many and hardly seems to try. It’s nice.
The Veracruz was Hyundai’s first attempt at a three-row crossover, nice enough in its conservative way, but it just didn’t sell. They discontinued it about a year ago. This year the Santa Fe has grown a third row, with the new Santa Fe Sport a shorter two-row model. If you ever find one, either the Tribeca or Veracruz are worth a look.
One car that OP can look for that meets all the criterion except Awd ( which is totally unnecessary) and gives him the newest and safest vehicle with the room suitable for a growing family without being a minivan, is 2008 plus, Scion xB.
I like the xB a lot, but have seen one that seated six. That is why his options are limited. Two patents, two sons who will be big enough to need adult-size seats in a few years, and a pair of grandparents who sometimes go places with them. It’s not like the old days of big cars with bench seats and column shifters. Not that anyone wanted to be stuck in the front middle spot. I guess some big trucks might have a front bench. Or maybe not.