Net economic benefit is not only GDP, it includes net tax over government spending. Immigrants are younger so they have more regular taxable income (vs capital gains), while spending more of it (vs saving) and drawing less government benefits.

The housing shortage is driven by NIMBY and regulatory capture. You are right that perpetual growth is unsustainable but we are not remotely near the possible carrying capacity of our land and resources. We have had periods of much greater population growth that were accompanied by greater prosperity overall.

Immigrants are not the problem, they are the convenient scapegoat of the business interests that have systematically hollowed out the middle class over the last 50 years.