They do a great job of hiding the fallacy by placing a single individual in the photo and saying "immigrant" instead of "immigrants".

I'm not sure about American law, but something tells me that it's even possible for an individual to do both things.

It's kinda hard to overwhelm the job market tho, at least if the economy is growing. In that case, more workers would likely facilitate growth, since the growing workforce will also have money to spend and create demand. If the market was already very competitive, things would be different, but developed countries are usually hungry for skilled workers. Demographics dictate this will only become more true as time passes