If Ned was genuinely concerned about Tommen & Myrcella's lives being threatened by Robert, he could have manned up and taken them under his protection. Half of Ned's downfall is his inability to stand up to those he loves, particularly Robert. His unwillingness to discomfit his friend is the source of too many of his moral compromises. While in the case of Daenerys, there was nothing he could really do aside from refuse to be a part of it, there were other options to protect Cersei's children, that did not require his giving her a headstart or motive to kill Robert.

The kids excuse only stretches so far. "I stole that money to feed my kids or pay for their medical bills" is almost acceptable. "I embezzled millions from my company to put my kids through private schools and give them vacations abroad" is not. Lying or fighting to defend your children against a murderer is one thing. Usurping a throne and betraying your king and country is a whole other kettle of fish.