The best approach is to use the law of diminishing returns.
To start with, the long view is that the problem is starting to resolve itself for three reasons. The first is that the Plan Puebla Panama resulted in a lot of Mexicans from southern Mexico being kicked off their land and told to go North, hopefully to the US. This was a nasty trick by the Mex government and caused a huge surge in illegal aliens.
But it is winding down. It was a “globalism” pipe dream and is pretty well defunct.
The second thing is that Mexican demographics. Their birthrate has plummeted to about 2.3 children per family, which is stability level. Without a growing population, there is much less impulse to migrate. Wages will have to go up, and their economy will do better.
Third is that the cartel war is making a pretty effective buffer, which when combined with negative publicity about the US, means that far fewer will be willing to make the effort.
The medium view is that erecting the wall on the half dozen major corridors used by illegal aliens will strongly inhibit crossing, as much of the border terrain is pretty impassable even in winter. So this is a very cost effective means to reducing the total number of crossers.
Putting up wall on the secondary corridors is a lot more expensive, but still eliminates a lot of crossers, so is still a good idea. Beyond that is extremely expensive and accomplishes little against common illegals, so is best done to keep out the drug smugglers.
After this, the question is *finally* how to deal with those already here. And this is where the law of diminishing returns comes into its own.
There is general agreement that the priority should be to boot out hardened criminals and international gangsters, like MS-13, the Mexican Mafia, and Cartel personnel. But there are so many of them that Mexico is begging us not to kick out so many, so fast, because as soon as they cross the border they turn into Cartel soldiers.
The way around this problem might be bureaucratic. That is, schedule them for deportation, but set up paperwork hurdles for their expulsion, and keep them in what amounts to high security prisons just north of the border.
This means they can be sent back to Mexico in an orderly way, flying them to Mexico City, so they are not just tossed into the arms of the Cartels, but are 1500 miles away.
Arizona, btw, uses this trick for “voluntary repatriations”, with free one-way flights to Mexico City for any illegal who wants them. Very few who take up the offer ever return to the US, and it is actually cheaper than the process of “just kicking them across the border”.
After getting out the really bad criminals, the next group are those who have not integrated and don’t want to integrate or become US citizens, but live in barrios up here and remain Mexicans, while sucking down government aid and health care.
These people are an expensive pain in the butt who need to be deported. Fortunately they cluster together in barrios like South Central LA, and their unwillingness to live around Americans makes them easier to round up.
Next, there are a number who are “good Mexicans”, want to become US citizens and integrate, are trying to work and not sponge, have lived in the US for many years, and have children who are US citizens.
These are excellent candidates to become US citizens, but right now are exploited for cheap, sub minimum wage labor, and often cheated out of their wages. A good solution for them is green cards, which will piss off the scum employers, as they will be covered by US labor laws. But they also need some means for in country application for citizenship, which right now is a nightmare.
This should not be free or cheap. Most are willing to shell out several thousand dollars, pay back taxes, and do a lot of hard work to get citizenship.
A subset of this group are their already legal children, who are US citizens by birth. Most of them are entirely integrated into the US, and are willing to support their parents efforts to become citizens. Very high quality immigrants even compared to those who have entered by using the cumbersome existing system.
So, put it all together, a more complex solution ends up with the US getting good, hard working immigrants, getting rid of violent criminal scum, ending the vast majority of future cross border illegal immigration, and getting new citizens in an orderly manner, with an annual quota, and not having to shell out a ton of money for welfare leeches.
What’s not to like?
As I don my asbestos Long Johns I would say that what you have sketched out is pretty close to what will have to happen. Motivated immigrants become inculcated within two generations.