I'm going to disagree with you here... This is age-old talking point of the democrats who have used this to try and justify "need" to flood this country with poor immigrants who they hope to recruit as the next generation of government-dependent democrats...
Firstly, American citizens who don't want these low paying jobs would likely change their minds if their government sponsored entitlements were to end... Disincentivizing lazy Americans will solve much of the demand for labor jobs from within...
Secondly, cheap foreign labor will no longer be so cheap if workers are actually documented, and employers have to pay taxes and insurance on them.
Cheap labor is not a viable economic solution... Flooding the workforce with cheap labor historically drives wages down across the board...
Covid was a perfect example of supply & demand in the workforce. Wages soared in labor-based sectors during and post covid, and these wages have not come down... This was especially evident in the hospitality and food service industry when American workers being paid more by our government not to work created a huge demand for workers that has yet to be adequately met... McDonald's is now paying $15-20 dollars/hour to start and still can't fill their demand for jobs. The problem is not a shortage of cheap foreign labor, nor will it be fixed by a deluge of more poor, dependent immigrants.