The crux of the immigration dilemma was nailed, unsurprisingly, by Milton Friedman and he summed it up in a letter he wrote in 2006, the year of his death.
There is no doubt that free and open immigration is the right policy in a libertarian state, but in a welfare state it is a different story: the supply of immigrants will become infinite. Your proposal that someone only be able to come for employment is a good one but it would not solve the problem completely. The real hitch is in denying social benefits to the immigrants who are here. That is very hard to do, much harder than you would think as we have found out in California.
Senator Barbara Boxer cannot, or will not, grasp this essential truth, and as Richard Simon reports in the Los Angeles Times:
... plans to push for Washington to provide $250 million and perhaps more to help local and state governments pay the cost of healthcare to uninsured immigrants who seek legal status under legislation now before the Senate.
A quarter of a billion is small money by Washington standards. Not enough to send up warning flags. But:
... Boxer, who is discussing her proposal with colleagues from both parties, said the amount could grow.
One does not have to be Milton Friedman to have seen that one coming.