No, they can't.
Out of the 23,388 wildly, astonishingly misunderstood things about immigration, this might be #1.
There's this prevailing idea that the illegals have this path towards work visas available but that they just choose not to take it. So 12 million people choose to live in the shadows, with all that that entails, out of sheer laziness, stupidity, impatience, love of illegality, or something along those lines. Well, I gotta tell you, that's not true.
THOSE WORK VISAS AREN'T AVAILABLE TO EVERYONE.
Those 130,000 or whatever visas divided by 100+ countries results is roughly 1,000 visas per year, per country. That's basically nothing so actually obtaining one is pretty much a lottery. They obviously run out immediately so 99.8% of people that would like to come here and work, cannot. So, many of them choose to do it illegally. Again, not because signing up for these work visas is too much of a hassle for them, but rather because it's their only choice.