OK. If you want to get technical, I will agree that asking for help is not disgraceful. But, two points:
1. You want to help people where many aren't even asking for it.
2. It is a different story for those on the other side who are giving or paying for the help (such as the taxpayers). It is their choice to help or not to help or how much help to give if they do, much the same way as if a parent decides to help or not or how much help to give. It is the decision of the givers, not the wannabe takers. Rich parents can sometimes be ridiculous in helping their children, maybe in buying them a brand new Corvette when they are sixteen to learn how to drive. Maybe by helping them escape to Mexico to avoid being put in jail for a probation violation. Other parents may help their kids at varying degrees. Some parents may have drug addicted children who they realize they have helped too much and the best course of action is to just say no and let the chips fall where they may. Other parents may have unemployed kids living at home into their 30's, leeching off of them because they are too lazy to work and the parents finally have had too much and finally kick their kids out on the street to fend for themselves. Sometimes tough love is what is needed to turn around these kids, instead of just giving them more, sometimes even when the parents are rich and could afford it. So, I like your analogy between society and individual families. In some cases the best thing to do is to not help in order to get rid of the dependency and cut the umbilical cord, even though there is enough money to keep people in dependency forever, generation after generation.