The policy referred to the signs is a legitimate one--don't feed the animals because feeding them leads to dependence. They are comparing this to feeding the welfare cases--pointing out that it leads to dependence. What they are missing is that not feeding them is worse--it leads to starvation.
(On the other hand, a better answer is aid that doesn't make it so hard to climb up.)
Very few recipients of SNAP or other assistance become 'dependent' on such aid. The overwhelming majority receive assistance for very short term.
An exception would be big business.
I feel pretty certain you actually know this but it doesn't suit your political bent to acknowledge it.
Simply looking at continuous time spent on welfare doesn't address the question. Most current recipients were already on it previously 1 or more times and after they leave it this time, most will return to it quickly. About half of current recipients have been on-off welfare previously for a total of over 60 months.
60 percent of people on it now who leave it wind up returning to it within 2 years.
Also, the people who are most likely to show dependency are precisely those predicted to become dependent by basic theories of learning. They are the people who either started on welfare before ever learning the habit of working (it must be learned), or people that have not worked recently and thus unlearned the habit.
Yes, there are many people that do come and go from welfare rolls quickly before they develop a learned dependency, and they are less likely to return because they strive to avoid the behavior that lead to returning to it. Again, that is just what learned dependency theory predicts for all animals, humans included. However, in addition to learned work habits, part of what motivates these people to get off of it quickly is a sense of shame to be on it in the first place. Another factor is the rather minimal amount of support they get on it that is insufficient for a decent quality of life.
This means that levels of dependency would increase if non-contingent stipends were increased, if the cultural sense of shame was decreased. It also means that dependency and future return the roles can be decreased by pushing those on it to return to work as rapidly as possible before the unlearn work habits, and setting higher limits on benefits given to young people with minimal work experience, reducing benefits to those that return too quickly, etc.. IOW, a combination of rules that limit the number of people that either never worked, or worked only briefly before starting or returning to the welfare roles, and also strict requirements of evidence that one is seeking employment and/or getting training, plus required public service activities for anyone on it for extended periods in order to rebuild and maintain work habits.
Also, there is clear evidence of trans-generational learned dependency that is beyond actual effects of real need. The longer one's parents are on it, the more likely the kids wind up on it, independent of actual SES factors that would contribute to such a correlation. Again, the most basic and established facts of human learning and socialization predict this would be the case.
Bottom line is that the easier it is to get, stay, and return to welfare and the higher the level of the stipends, the less hard some people will strive to get off and stay off of it. Every valid behavioral theory and piece of empirical data support this. Welfare is not a fun grand 'ol time. But finding work, working, and maintaining a job are not fun either, so anything that makes welfare more appealing or just less unappealing in relative terms will reduce people's efforts to find and maintain work and thus stay off the welfare roles.