Rent (and the big real estate problems) are because separate awful restrictions on building more properties. That's a local problem, created by NIMBY-ism of locals, not immigrants.

We mix that with immigration, but there's no reason to. Typically workers _love_ new businesses around -- more work, more business.

Rent is mixed with immigration because housing (extending that, physical space) is a limited resource.

If housing is static and never increases, no matter how much the Average Joe's pay is increased due to an influx of highly paid immigrants, they're still going to lose when competing against a highly skilled immigrant for limited resources.

In reality housing is not static, but as you mentioned is highly regulated. The established wealth have incentive to not increase housing supply, especially if they have a number of properties in an area with a lot of immigrants (one such source of NIMBYism).