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.
[1] https://www.axios.com/2024/04/25/trump-biden-americans-illeg...
[2] https://www.nationalmortgagenews.com/list/homebuilders-see-p...
[3] https://finimize.com/content/pultegroup-profits-on-housing-s...