There's no observable nor identifiable lack of jobs that's occurring because of labor performed by illegal immigrants. The vast majority of work they undertake is work that American workers simply won't do.
What you're describing is Myth #2 that's explained in more detail here;
https://www.pbs.org/newshour/economy...he-u-s-economy
Worse, this is a factor that only really exists
because of restricted immigration. If they were granted easily-accessible work visas and thus enjoyed worker protections, the ability for employers to undercut minimum wages by paying undocumented labour under the table would be severely restricted, compared to the current status quo.
Yes. Your thesis
does need defending. Because it's absolute hogwash.