The Constitution vests sole authority on immigration to the US Congress.
The authorities have the guns, they can and do impose their will, not just on immigrants.
I'm not going to go there, but again- a wall is an object. It is not a policy. A wall (or lack of one) does not affect immigration policy. There is a great deal of trafficking in both drugs and humans at our southern border, and the wall is intended to help address that issue.
The CBP are the best ones to ask where and how much wall is needed in any particular sector, and it is always brought up in the hearings. You can find those hearings in the C-Span archive if you are so inclined.
Same with border apprehensions over time, the CBP publishes the numbers. But those numbers require a lot of analysis to extract meaningful information, because policy is the big driver, and the definition of "apprehensions" is fluid.
Understand that there is immigration and there is illegal immigration. I don't know any country where someone can just enter without checking in first- at least not one that I would want to visit.
Immigration is good. Illegal immigration is not good. That isn't a radical viewpoint- I'm pretty sure your authorities look at it the same way?