The median net worth in the United States is approximately zero (assets less student loans, mortgage debt, etc). The median 401k balance in the US is something like 100k. I'm probably right of many people here politically but will readily admit, there's a lot less wealth out there than you'd think, and what there is, is held by a pretty small number of people.
Also, speaking broadly, I don't think the economics profession really understands the connection between house prices and rents. It's a complex topic with a lot of conflicting information. I definitely wouldn't take it as an article of faith that more expensive housing necessarily implies higher rents. There are all kinds of complex subsidies like mortgage interest deductions, factors like credit availability, short-term fluctuations in material and labor prices, etc that make a straight-through 1:1 correlation too clean.
As a general comment, it helps to disaggregate when thinking about huge topics like rental inflation. Rent is growing fast in western/mountain markets (e.g. Idaho) and places like Miami, while hardly budging in places like Cleveland or St. Louis. I'm sorry not to have citations on a lot of this, it mostly comes from firsthand experience and a lot of reading -- I read the economist cover-to-cover every week, manage 12 rental units, and talk to friends and family spread across the US (Seattle, DC, Chicago, Indiana, etc) almost every week.
