The obvious problem with ruthlessly enforcing E-Verify is that most people, in their day-to-day reality, like consensual exchanges, including with illegal immigrants. What people don't like are videos of undocumented Guatemalans setting subway passengers on fire, or caravans flooding across the border. They want a sense that things are “under control.”
The distinction here can be thought of as daytime and nighttime preferences. At night when watching the news, many people are worried about illegal immigration—this is the realm of abstraction, where pixels of a woman burned alive by a Guatemalan migrant grace TV screens like shadows on the walls of Plato’s cave. Here, law and order prevail. But in the day-to-day, people go back to behaving like mostly rational economic agents, entering consensual exchanges with undocumented Guatemalans. The correct equilibrium if you are a skilled politician, of course, is to pander to the former while preserving the latter. Results may vary depending on the immigrant population in question, but suffice it to say, this isn’t so difficult in Texas.