Texas is a border state, and its economy is doing just fine - much better than California's. This is due to the level of socialism that exists in each state rather than them being border states.
2. Millions have crossed. The mitigation has been very unsuccessful. It's arbitrary conjecture to speculate about the volume of immigration otherwise.
3. Your admission that "they're obviously not actually trying to control the border" drives my point home further. Passing more legislation will not be effective at mitigating the level of immigration. How many times must government prove that it's rather useless at achieving intended goals?
4. You're mistaken if you think there is a limit on the size of the economy. That is the same argument used against technology. I suggest reading Henry Hazlitt's Economics in One Lesson.
Socialism has already taken hold, if you haven't noticed. In the OP, I included a provision in my proposal that incoming immigrants not be allowed to vote or receive welfare benefits. Further, the more people that siphon off the public trough, the sooner socialism meets its demise. Ask a Russian. There becomes no incentive to work. In addition, there's no guarantee that some of the immigrants won't be as influential as Ayn Rand or the like. Not all immigrants are anti-liberty. Many are probably readily aware of socialism and come to the US for more opportunity rather than to recreate the environment in their country of origin.
I am analyzing the the economics with long-term political consequences in mind while you seem to be weighting your analysis with short-term political consequences (a Keynesian trap).