Ease of car use is directly opposed to ease of transit/walkability/bikability. You can't have good transit/walkability/bikability with ubiquitous parking and car dependent infrastructure. So you're setting up an absurd hypothetical where driving was easy AND walking, transit, and biking were also easy THEN the rich would still use their car.

But those things can't coexist. Car infrastructure and culture is actively hostile and disruptive to all other forms of transportation. Good transit and walkability necessitates it being more difficult to drive.

So if anything you've just changed the nature of the second order causation from

Transit -> More housing -> less emissions

To

Transit -> less driving -> less emissions.

It's still causation.