It shouldn't take far-seeing, risk taking, and great inconvenience to get out of poverty. If our society wants to bring people out of poverty, it should make the choices leading to that obvious, low risk, and convenient.
If the context was game development, this wouldn't even be an issue. People do what they're incentivized to do.
Also your conclusion about "rational decision" is clearly wrong. Taking risks is not necessarily rational. Doubly not when you are a single mother with 3 kids and a deadbeat ex.
Taking the job guaranteed losing her money. If she kept the job for a couple of years, she would have started to break even, but it would be quite a few years until she was ahead of where she was. Instead she continued to look for a job that either paid just little enough to keep food stamps, or offered a prospect of paying enough to offset losing food stamps.
Your reasoning is based on the theory that your best way up is to take the best job you can and then work your way up. That was how things worked when I was a kid. But today people tend to improve their job by switching jobs, not by being promoted in their current one. Therefore staying underpaid is often a better route up than switching to a job you don't want in the hope of maybe being promoted some day.
That being said I do agree that if there's a huge difference in future earning power, it would still generally worth it to go for the job that involves giving up food stamps.
If you can manage to hold your life together well enough to continue to do the job, with fewer resources available.
One of the main problems with the existing social net is that it comes with "economic cliffs" where making a little more is not to your economic advantage. And then people who make the rational decision are seen as "lazy".
For a real example, my niece had to turn down a job because it paid her $0.20 too much to stay on food stamps. And she couldn't be paid less because it would have broken union rules. So she did the rational thing and turned down the better job. Which meant that her skills and education went underutilized.
One of the big arguments for UBI is that this kind of thing doesn't happen.