I would imagine more young, ambitious working age adults would help the welfare system, not hurt it.

If you look globally at countries which have issues with their large social services, they're almost all mostly homogeneous and declining in population, especially among the young. Which makes sense if you sit back and think about what social services are typically offered and where the money comes from.

Friedman's argument was more so to just keep them as illegals but not deport them. That way they can support the welfare system but not use it. Friedman didn't want to make them legal until the welfare system has been crushed.

Of course that might require some changes to make it actually true illegals don't use state benefits. You need to cut off WIC for illegals, public schooling for illegals for instance before they will actually not be using public benefit. Also their children become legal via jus soli.

The obvious down-side is that those citizens / legal residents who have the skill level of illegal immigrants (sad, but commonly true) will see their real wages depressed and more competition for the job.