I do think that once an elderly person loses the ability to drive, it's often a big tipping point towards their decline. I would suspect that losing the ability to drive usually (but not necessarily) comes before losing the ability to navigate public transit.

But I don't immediately believe the link that 'car culture' -> 'earlier cognitive decline'. Car culture, for example, is usually associated with living on larger plots of land, which comes with its own set of tasks and chores that can keep someone older occupied. A smaller apartment requires much less ongoing work.

I think a lot depends on the individual and how they best stay active. More dense living probably provides easier opportunities to do things, whereas less dense living sort of forces you to perform ongoing tasks.

If the additional area is used for new hobbies, hosting guests, or something fulfilling and interesting, then sure it can help keep the mind active. But people don't usually retire so they have more time for vacuuming or dusting. Many elderly people simply don't have the energy or interest to maintain their homes, it slowly falls into disrepair.