Perhaps the confusion arises because the immigrant does not necessarily purchase goods and services produced in their new country. We all work in a global economy were its impossible to follow all of the transactions between what we produce and what we consume. That would be true even if nothing ever crossed international borders.
For example even if the number of mouths to feed increases, the number of farmers cannot increase beyond what there is farmland for. So the hands will be competing for who gets to plough the fields, in exchange for a meal. Too many hands along with too little land, and the losers starve.
And, it is not necessary that we would just raise salaries without cheaper labor available through immigration. Many jobs at the lower end would simply be automated or a service might disappear altogether (e.g. gardeners are a luxury, too expensive you just do it yourself).
I hope someday we'll be able to think more globally, maybe we can in a future post scarcity economy.
That's from a purely economic perspective. What nobody ever wants to discuss is that creating an ever-more culturally and ethnically fractious country is a huge problem looking ahead decades quite aside from near term GNP prints.