tldr: I'll be in favor of cracking down on illegal immigrants when people stop hiring them for fear of getting in trouble.

This is a naive reading of the politics. Very few people are actively in favor of illegal immigration. Many people believe the cures (that is, the ones that politicians push for) are worse than the disease. My personal view is that if we really wanted to solve the problem, we would do it on the economic side; there are whole sectors of our economy that everyone knows are propped up by the employment of illegal immigrants. Nobody talks about that because the people who benefit most are strong, organized, and relatable (upstanding well-to-do businesspeople). Instead, everyone goes after the weak, disorganized, and unrelatable (scary foreigners who don't even speak our language).