Like I said long ago, it sounds to me like your flight was combined with another. In essence "those people" might have already had their seats on that flight--so now because your flight no longer exists, you think it's right to bump them out of seats they had reserved?
If you don't want your children to sit alone, you need to first off get to the airport very early. Like around four hours early. That's when they start to release the unused frequent flier seats. If you still can't get seats together, then get to the gate and ask again. And let them know that you are willing to take a later flight if they can't accomodate you. They will probably ask people to move, but will not force them. If they can't accomodate you, it's my experience that they will move you to the next flight that they can accomodate you on at no charge, but it might not be that day, and they will not give you any expenses for the delay.
It is NOT the carriers fault, as you recognize, your TA is a moron, and they shuld do whatever they can to take care of you.
But I've got one thing to ask...what if (and this HAS happened to me more than a few times) you get the seats together and your flight is cancelled due to weather? And the only option to get you home for the next three days is to take seats all over the plane?
In the case of weather related cancellations and delays, they are under NO obligation to pay for any of your expenses due to the delay or accomodate you in any way other than offering available seats on the next flight--which will probably not mean "together."
And out of curiousity where are you flying from?
Anne