However, fresh immigration does drive down the pay for trade work in areas where immigrants are skilled

Let’s just be realistic about one thing, there will always be fresh immigration and you will never see all 20million undocumented people deported.

The problem is that people who told you to be worried about immigrants are using your fear to keep them as an ‘illegal’ underclass. If they were just accepted as human beings and given the same legal rights and protections as everyone else, they too would have to be paid above minimum wage. They couldn’t be forced to work crazy hours for fear of deportation. You would have more people able to join your unions and make your attempts at collective bargaining stronger.

So the answer to your concerns, if genuine, is to stop demonizing these people and start the process of giving them all documentation.