They still hold a decent amount of cash as a percentage of their portfolio. 10% of a million dollar portfolio is still $100k which is more than the average joe has. He would use that money to buy a house instead of letting it sit around.

Yes, bigger down payments suck but that is mostly a zoning/housing supply issue.

I’d guess that many (perhaps most) wealthy households hold far more dollar-denominated debt in their real-estate and business interests than they do cash.

It's very rare for wealthy people to hold 10% of their assets in cash (or even cash equivalents). When they purchase real estate they usually take out loans secured by their other assets.