Ok, this may not be the most popular opinion, but I have thick skin and can take it.
Maybe the dogs thought they were trying to protect your kids from you. As has been pointed out, dogs sometimes view children as other dogs. Especially those who are part of a family that has kids.
According to your account, they did not show aggression until you put yourself between your kids and the dogs and assumed an aggressive posture. Dogs can sense and are sometimes very responsive to human emotions. You already stated that you were afraid during the encounter. You also had quite a bit of adrenaline flowing. The fact that you vomited after is a good indication of that. Dogs can smell that. I am pretty sure that your wife was giving off a similar scent.
So, look at it from this perspective. The dogs see your kids as potential pack members and now they have two large predators (cuz that is how they view you) who are surrounding them and smell like aggression. There is a good chance that they were not looking at your kids at all. They were keeping tabs on your wife. In their eyes they were looking for a way to separate you and your wife (the aggressors) from potential pack-mates (your kids).
That would also explain the behavior once your wife and kids were in the house. Your relief at that causes your fear to subside. The "aggressive" smell declines and the potential pack mates are out of sight. Time to move on.
Now, I am not saying you didn't handle the situation correctly. They very well could have had ill intentions and would have attacked if provoked. Either way, the situation came out positive, so the actions to get there don't really matter in hindsight.
That being said, there is a good reason why "dog people" don't get attacked as much as "non-dog people". That is because us dog people will usually greet strange dogs with caution, but not fear or aggression. Once you start to exude fear or aggression, the dogs will react in kind.
So, I guess the point of all this is an adaptation of something that Bear44 posted here not too long ago: Meet every dog with cautious respect, but have a plan to kill them if necessary.
Wrong read on this. They had no reason to believe the kids were part of their pack. They had never met them before. The reason they became aggressive when the OP put himself between them and his kids is because they were ready to fight for their dinner. Exactly the same behavior you'll see in nature when an adult prey animal tries to protect their young from wolves.
When the kids got in the house the prey was no longer available. Time to move on.
