Isn't this more about supporting companies' wishes to employ illegal immigrants at salaries which no US citizen would work for?

Whether they have jobs or not they're still illegal immigrants. In my view this is rewarding people for breaking the law and it sets the wrong example to those who obey the laws. It is not fair to the immigrant that tries to get into country the legal way.

In theory it is being said that they are taking jobs that Americans don't want but this proposal would promote more and more Americans to live off of the system instead of finding a job since the pay scale would be driven down lower and lower.

I'm not sure what exactly the solution would be to an immigration problem but I would think that the laws currently in place would be inforced until they were changed.