You are both correct. Immigration does increase GDP, and wages are also stagnant. The problem is that the GDP gains created by both immigrants and native born workers have concentrated in the holdings of the capitalist class (to which even many upper middle class Americans belong in varying degrees), because of low taxes and asset price inflation.

As a result, public investment funded via taxes has moved to deficit spending and sporadic and selective philanthropy, exacerbating both the inequality of wealth and opportunity.