It sounds like the problem here wouldn't be the immigration, but rather that moving between economic classes is difficult for those with no skills. If we could provide ways for any citizen or immigrant to gain useful skills that are in demand, then we would help mitigate the problems of unemployment as a whole and help reduce the spending burdens of social programs.

The reason he has a depressed wage is because he's undocumented and therefore can be exploited. Reduce the friction involved and more people will be documented, and more on their way to full citizenship. Shady employers won't be able to pay less than min/market wage since the now-documented workers have legal recourse.