That's a great comment, except for the last bit. The evidence does not support that welfare grows the welfare population. Concur on the other issues, though. London, where we live, is similar.
I no longer recall the title of the book but I read the history of this in the U.S. I did not say welfare grows the welfare class. It grows the number of single mothers who, thus, live in poverty. When welfare was designed in the U.S., "poor, single moms" were mostly widows and considered "deserving poor." They did everything right -- got married, had kids within the bonds of marriage, etc -- and had something unfortunate happen.
In Europe, a lot of "welfare" type programs are designed to help women or pregnant women or children or families regardless of income level or marital status. In the U.S., such aid is almost always tied to some criteria proving you "need" it. So you can't get help until a) you already screwed up and b) you are willing to fill out forms testifying that you are a screw up. This has substantial negative psychological impacts that I don't think you see in Europe (yes, I lived in Europe for a time and have read books and articles comparing European policies to American ones in this area).
In the U.S., welfare -- which requires a woman to be both a mother and unmarried to qualify -- actively discourages "shotgun weddings" and changed the social contract so that having babies out of wedlock is now much more acceptable than it was when the system was conceived. At the time, it was inconceivable that women would choose to intentionally have babies out of wedlock. This is no longer true in the U.S.
So, that is how welfare in the U.S. grew the population of "poor, single moms" -- by actively encouraging out-of-wedlock births. Single moms are typically poor. Families with two parents are usually better off.