@mdk The likelyhood is that living off grid ect is better than what they fled. If that weren't the case people wouldn't keep coming. A sane and moral response is increasing efforts to address the problem down stream. There aren't easy answers to this problem which is what the build a wall and deport them all crowd refuses to see.
A country has the right to control its borders and set its own immigration policy but what is legal and what is moral are not always the same thing.
It would certainly make me happier if there were a reasonable legal way for people in these terrible situations to find a measure of sanctuary.
I mean there is (kinda -- that system is also pretty broken)... but it's so much easier to come illegally and, like you say, it's not like these poor folks have time to sit around. As long as illegal immigration remains this easy, effective reform is impossible. The path of least resistance is, and will remain, to come here illegally -- until we have a physical barrier. Without that, we're neglecting a huge, huge source of poverty, crime, and other badness.
Speaking of which -- it's just as easy for the drug lords to get here as it is for the people fleeing them. MS13 is worth googling, if you're unfamiliar with them (dunno how international their reputation is). We have to do something here -- flinging wide the gate and crossing your fingers feels very nice, but it's terrible for everyone in the long run. I say that to do nothing to secure the border is neglect.