No, you're analogy is still incorrect. Even if one is a minor aspect of a broader activity, they are still completely different aspects. Breaking and entering followed by shoplifting (which is just theft) might happen together but they are still different crimes. Someone can commit one of those crimes without committing both. You can illegally enter the U.S. and never take a job, use benefits, etc. Just live illegally. You can enter legally, overstay your visa and work illegally. 2 separate crimes require 2 separate responses. You cannot treat them as the same unless you ignore the actual crimes themselves.
Second, a wall doesn't mean closing the border. The hypo was what's wrong with a wall, money aside. That is a different question from "should we close the border?"
So, again, shoplifting requires actually taking something, breaking and entering only requires entry. Conflating the two in your analogy is simply misapplying the definition of well-defined terms. It's far easier to just re-write your point using the proper crimes - illegal trespassing or some other crime specific to illegal entry, like burglary - which is illegal entry with the intent to commit a crime.
Why you would insist on the wrong language is baffling to me. You could still make your point...unless you don't think your point stands up when presented in terms of people breaking into other people's stores.