There are jobs that pay minimum wage - generally unskilled jobs, especially in retail. Food service is another example. However if one doesn't rule out manual labor or jobs that require some level of skill there are jobs that pay significantly more. Unemployment offices generally (here in NJ at least) still provide some level of job training, not for the most pleasant or high paying jobs but jobs that pay 2-3 times minimum wage. The problem is people don't want a job that requires heavy manual labor, or so it seems. Many people will take these types of jobs in the spring or fall but bail out of them when winter comes around here in the NE/mid-atlantic areas because of the cold. They opt in many cases for unemployment. Unemployment was extended from 26 weeks to 99 weeks. Some states are finally starting to roll this back to 26 weeks (mainly because the states are going broke supporting these people), but the damage is done.
I would rather see our unemployment / job services departments work to find the type of jobs that people will enjoy and excel at instead of sending them out on their own to fail. That's not what we have in this country. We have people that do not necessarily know what they are or would be good at doing jobs far below their talent level in many cases and convincing themselves they can't do any better.
Even worse we have the trap you described well of welfare - hard to get out of once you are in it. The majority of those using our welfare system are single parent households where there is no option for one parent to work while the other raises the children. In addition many of these people have multiple children that they simply cannot afford. How about putting some restrictions on this? If you are on welfare and you have additional children, no more welfare. If you cannot support your children, they are placed into foster care until you can. Pitch that and the liberals in this country will crucify you. Instead we just let these people sit at home and collect the money that the mere 47% of Americans that pay federal income taxes continue to pay into the system. And Obama wants us to pay more.