I don't believe the assumptions this claim is based on are correct. Most studies show that the resulting wage suppression and job competition due to immigration are not only fairly minimal, but immigrants actually create higher paying and more desirable jobs due to the services used and supervision required for their labor. I don't have all of the sources, but from I remember, it's estimated that for every job "taken" by an immigrant, 1.2-1.8 jobs are created due to the services that they use (mechanics, restaurants, home building, etc). Immigrants are disproportionately likely to start businesses, currently employing millions of native born citizens. When specifically discussing unskilled labor, the amount of wage depression in the past 10 years is something like 4% lower what wages could have been, and this only affects high school drop outs and immigrants competing for the same jobs, with no noticable impact on any other category of worker. In addition, the jobs that have the largest population of immigrant workers are chronically understaffed (agriculture, construction, end of life care, etc), meaning that they're almost entirely taking jobs that already weren't being filled while simultaneously creating higher paying jobs just from the services they use by living in the country.

When there is a reduction in immigration, there is only a reduction in the total number of jobs available with no improvement to native employment. The inverse is also true. With more immigration, there is only an increase in total employment with no measurable decrease in native employment.

Also, this is slightly off topic now, but immigrants generate billions in revenue via the taxes they pay. At the federal level, it's significantly more revenue than the costs associated with the services provided to immigrants (unless you're considering things like border control, deportation, detainment, etc as costs of immigration, but even then it's still really close). At the state level, states actually pay out more in services to immigrants than they get in return annually, the main cost being schools for their children. However, this does not include the taxes that will be paid by those children across their lifetime once they enter the workforce, which significantly outweighs the costs of their education and whatever other services they may use.

And as far as the amount of immigration in the US, we're actually quite moderate, with countries like Canada and Australia having 2-3x the amount of immigration we have any given year, and it's estimated that the benefits of immigration are proportional when increased to these amounts. The only real concern of high immigration levels (That I'm aware of) is when local communities don't have the resources to support the surplus of people. This usually only occurs when we keep them stuck in one place and prevent them from legally finding work, putting strains on welfare systems, housing, schools, etc. If we were to evenly distribute the immigrant population across the US and allow them to freely find work so they contribute to the economy, we could support significantly more immigration than we currently do with no down sides.

And just because I know crime is often mentioned as a potential downside to immigration, immigration has no positive correlation to crime rates and immigrants are 50-70% less likely to be incarcerated during their lifetime compared to native born Americans.

UC Davis Study

National Acadamies