Yours is a recipe for inflation. The shelter thing is key. We need to build a lot more of it at a price we can afford and that means we have to keep a lid on wages, which will be OK because we'll be driving down the cost of shelter. Fully agreed on the reduce profits bit; middleman profits are the key component of both shelter and grocery inflation.
Shelter inflation is driven by supply shortages. There is no appetite for increased immigration crowding out domestic workers to keep prices low for new inventory homebuyers. The electorate already has substantial support for deporting the ~11M unauthorized folks in the country today [1]. Mass deportations are highly unlikely, but it is a reasonable proxy for increased immigration appetite. Build more housing, pay living wages to labor performing the work, cut cost of capital, compress profits for new housing if necessary [2] [3]. As your comment mentions above with regards to the credit card spread and profits, homebuilders are milking the housing market in a similar manner, because they can. More immigration doesn't solve that, it just makes us all worse off (reducing domestic worker opportunity and increasing affordable housing competition) to support homebuilder profits.
