> came to work jobs no one else would take
This is a particular piece of rhetoric I find distasteful. If there are jobs that are structured such that only an illegal immigrant, who by definition exists outside the normal labor pool and its protections, will take, that job should not exist in that form or at that pay rate. Full stop.
So the fact that jobs that "no one else would take" exist and are being taken by illegal immigrants isn't a good thing. If you're in favor of expanded immigration, making that claim doesn't help your case. It substantiates the case that anti-immigration proponents have always made, which is that immigrants take the bottom out of the labor market and help keep salaries down.
After all, if nobody would take a job doing some particularly onerous job at a pittance per hour, then the employer would have to pay more, or automate, or find a more efficient way of doing the job. Refusing to do these jobs is the correct response when they are clearly undercompensated.
There are lots of people in the legal labor market who do terribly unpleasant, physically strenuous, or frankly dangerous jobs, but they typically (outside of illegal or exploitative markets) do them for reasonable wages. Someone who SCUBA dives in raw sewage or nuclear waste, for instance, is probably going to demand a fair compensation for the unpleasantness of the job. The same should be true with people who work in slaughterhouses or picking strawberries or tarring asphalt roofs. The narrative that "Americans just won't do" certain jobs is one that is created by cheapskate, exploitative employers who don't want to pay the market rate for particularly ugly jobs. Parroting it is water-carrying for these exploitative industries.