> What's the best alternative way to minimize the number of people in the country illegally?

Fix the rules on allocating legal slots to be sensible, starting with aligning supply with demand by eliminating the per country proportion limit of the global allocation in each category would be the best alternative, but you don't need the best to be a better alternative.

> It's fine enough to say this option sucks, but you have to bring a better one.

Literally doing nothing would be better, on a net benefit after costs perspective, then building a wall.