Okay then, Lexikon. Let me put it this way.
Even if the government re-training paid for all of their food, their needs, and their bills, you would then deny them welfare if they don't get a high-paying job...
...which isn't always in their control. For one thing, corporations don't WANT to pay people high-wage. They want to pay people as shit wages as possible for as much work they can get out of them as possible.
And considering that unemployed > potential jobs, the corporations can afford to pick the cheapest bidder. They don't care about actual skill, just so long as they're qualified.
By your logic, the people who don't get picked for arbitrary reasons then lose welfare even if they keep looking for jobs but don't get them.
Your logic is exceedingly dangerous. Welfare is a safety net. It shouldn't have the qualifier of "You only get welfare if you make yourself useful to society" because that is a terrible, terrible line of thought. It's a safety net. Its only requirement should be when you're too poor to sustain yourself. None other.