The current system will not change soon, because there is a powerful coalition of interests for more and more immigration, preferably of illegals and the poor, from both of our major parties.
It suits a majority of businessmen to have a large supply of cheap* labor, preferably of illegal immigrants because they have less bargaining power than legal immigrants or natives, and these businessmen transmit their preferences to the (usually) Republican politicians most sympathetic to them. It suits Democratic politicians directly to admit a large number of poor and uneducated immigrants, because these immigrants generate future Democratic voters. Examples: California used to be Republican. Orange County was a bastion of Republicanism.
Both sides recognize that the resulting mockery of law offends a significant number of voters. So they support ineffective measures to limit illegal immigration, such as border fences, or they set themselves up in opposition to ridiculously easy amnesty proposals. And most of the voters fall for it.
*Cheap as measured by the direct costs borne by the employer. Adding in the cost of educating the illegal immigrants’ children, subsidizing their rents, their uninsured accidents etc, they may not be cheap. But those costs are borne by taxpayers in general, not directly by the hiring employer, so the illegal immigrants appear to be cheap. Illegal immigration is a taxpayer subsidy for employers and a growth source for the Democratic party.