@gollum,
I think you would create a false barrier and some perverse incentives with that plan. The current system puts all the risk on the workers. Shifting it to the employers would probably result in less available workers since employers are notoriously risk averse. Estimates say there are between 11 and 12 million undocumented people in the US and currently we have pretty much full employment. The US Chamber of Commerce
shows labor shortages all across the economy. If those millions of workers disappeared tomorrow, the economy would absolutely crash.
If they were all legalized, the economy would likely boom since there are significant costs to maintaining a shadow workforce, but the reality is that stoking hatred of immigrants is a winning political strategy in some places so we aren't going to see a common sense approach any time soon.