On one hand they're politicians, so "thinking things through well enough" isn't something they're really all that good at. On the other, ask yourself whether the long game is really "to protect the children", or whether it's something deeper.
I understand what you are saying, there are legitimate pro's and con's on both sides to this issue. I tend to lean on the cautionary side. So I think implementing a strict background check, and keeping a state to state database to background checking for the serious mentally ill who have been involuntary committed to a phcych hospital is necessary. I also think if you have a family member like Adam Lanza's mom who is known to suffer from a serious mental health issue that the family should not be allowed by law to give access to their guns, to that family member, and must take appropriate steps to ensure they can't get to them. If someone dies because they did not, then they get jail time and lose the right to own fire arms.
There are solutions but I think the republican base and the NRA have done a good job of scaring people particularly gun owners and misinforming folks on what is trying to be done. They aren't taking peoples fire arms away they already said that.
Now if the states want to do a state to state ban on certain assault rifles, well then that is a state thing. But as far as the federal government, I am definitely in favor of a strong back ground check for these things which can be tapped into a national database and easily accessed by law enforcement and gun sellers.
Anyway, I know so many here disagree but if you think about it, most of the whacked mass murderers by gun shootings have been mostly middle class white males who had a severe mental illness.
At least we can try to keep guns out of their hands. There really isn't much more that we can do really. But we can't just not do anything, simply because its a worthless effort. We must try to do something, reasonably.