I don't have hard numbers but anecdotally it's a lot of people. If you listen carefully you'll hear a lot of things like "Renting is throwing money away", "I want to get on the property ladder", "buy land because they aren't making any more of it", etc.
The American Dream™ centers heavily around home ownership and the government subsidizes it by allowing mortgage interest deductions and underwriting zero-down loans. Getting a mortgage and building equity is the only feasible path to wealth for working-class people who don't have money lying around to invest or the financial literacy to start doing so [1]. Contrast this with reality: in the market I live in it's actually around 20% cheaper to rent on average (presumably because California caps property tax increases). For 30 year mortgages in a sideways market you generally have to make payments for 5 years before you can break even when you sell. On top of that houses are non-fungible, illiquid assets that require maintenance and are subject to black swan events like termites, natural disasters, and destructive tenants that might not be covered by insurance. On a rational basis they are mediocre investments but our culture has conditioned people to think this way because it's their only hope for a nest egg. One might also recall what happened circa 2008 when banks encouraged the house-poor to leverage themselves to the hilt via various home equity loans to the delight of sports car and big screen TV manufacturers.
I don't think this is a healthy state of affairs but these are the breaks.
[1] I didn't gain a solid understanding of personal finance until after I'd already bought a house with nothing more than several thousand dollars in my checking account - stumbling across Bogleheads via the web saved me.