Establish "Ellis Island" type facilities to vet and either accept or reject, and put immigrants on a productive path to citizenship that tracks their progress. Every year all report their status, with help of us, the American people, to ensure they are progessing. Just like it was done in the 20th Century up until Kennedy and Johnson. Every January, media had public info spots about the alien status reporting mandate, asking all Americans to help. Game shows ended with these spots. It worked.
Their addition to the legal work force reduces stress on American retirement and health entitlements Social Security and Medicare, so their flex won't break backs of workers, and will be there when THIS workforce retires. Just the way it was designed.