My problem with your analysis is the same problem that I have with over-crowded prisons that give early release to felons. Outlaws are outlaws and shouldn't be given a pass.
My remedy? Bring back our troops from the Balkans station them on the border. Beef up and expand the INS. Mine the Mexican border. Put up guard towers. It's all a part of recognizing that there's a problem and dealing with it.
That sounds like a good idea, but you realize that it wouldn't be enough to stop illegal immigration. Even if we build a 20-foot wall across the border and post soldiers every few miles -- which would be prohibitively expensive - some illegals would still find ways to get through.
As long as the standard of living in Mexico is that of a third world country, we will always see desperately poor people trying to cross the border seeking work or welfare. A family on welfare in the U.S. enjoys a higher standard of living than millions of working poor in Mexico. Wouldn't you do the same thing in their situation?
We don't have an immigration problem with Canada because Canadians enjoy a reasonably good living standard of living. Also, Canadians who can't or don't want to work won't come here for welfare benefits since they're higher in Canada.
Your proposed solution means more expenditures and bigger government and it will not completely eliminate the problem. Instead we could simply limit welfare to U.S. citizens or permanent residents. That would greatly reduce government welfare spending, which means smaller government while removing the incentive for aliens seeking welfare to enter the country illegally. We'd still get illegal aliens coming here to work, but those aliens add to our economy by providing a valuable commodity (labor) at a lower cost than American employees would accept. OTOH, the illegal aliens who contribute nothing to our economy would no longer have an incentive to drain our welfare rolls.