I am not familiar with the attrition method.
This is whether you like it or not how Americans feel. They do not like the chaos in the border, having to pay for illegal immigrant healthcare, education, crime as well as the obvious national security concerns with uncontrolled immigration. On the other hand many know people who are and have been here illegally for a long time, have built ties with the community , have AMERICAN born children as it is the LAW now and work decent jobs. Deporting those people either forcefully or by denying them the chance of working (end result, the same) just does not look or feel right. There is where the American soul IS. The longer the GOP hardliners fail to come to terms with that, the more electoral peril they will face from Hispanics and non Hispanics alike.
Step-up enforcement against those using phony IDs and illegally claiming rights which belong to American citizens-- public benefits, voting and the like.
As certain milestones are met, issue, at each increment, a limited number of green cards to the most productive of the illegals . . . based on demonstated willingness to stay off the government teat, pay taxes, cooperate with law enfocement and learn English.
Those who might otherwise qualify but are over the mandated maximum for the increment could be issued an extension of say, 6 months, and get a provisional green card.
As demonstrated enforcement continues, provisional green cards could be upgraded or extended. The freeloaders, gangbangers and most undesireable element would even self-deport once enfocement action appeared serious.
In short, there would have to be a stick (demonstated enforcement action and progress) to get the carrot (additional green cards and extensions of the program).
And one more thing-- the ethnic lobby would not like this, but cooperation of their country of origin would be one of the factors considered in who gets preference for green cards. Support us, cooperate in extraditing criminals for crimes committed and encourage the people you send to behave as guests . . . move to the front of the line. Tepid support, no cooperation and encourage the people you send to behave as colonists and freeloaders . . . move to the back of the line.
Attrition is also known broadly as Enforcement-First. But basically it means you combine serious border-security measures with serious interior enforcement so that the influx of future illegals is greatly reduced, and the current population of illegals is gradually reduced. Tools include walls and fencing, more border patrol agents, employment enforcement, cutting off of drivers licenses, non-emergency public services, and in-state tuition, and yes, deportation too (though it could be focused on criminal aliens).
Maybe if we tried this (and I mean really tried it), it wouldn't work. Who knows? But its certainly preferable to try first over the path to citizenship approach, which we know would result in many more Democrats in the future.