As a general comment, I'd like to say that getting out of the house is a hell of a lot easier when you don't have to drive everywhere to participate in daily life. So and so family member sits at home and watches TV all day is a phenomenon caused primarily by our car-centric culture which, for the elderly, is a barrier to staying healthy both mentally and physically.
I'm no fan of car culture but I think to say it's the primary cause of living a sedentary life at home is an overstatement. I deliberately moved somewhere where I could walk to everything I needed including a fantastic central train station, I no longer even own a car, and yet... over the years my habits changed and I now spend a ton of time at home. My motivation to go out has simply declined.
There is a relevant concept in psychology called activation energy, James Clear provides a good introduction to it. Certainly in recent years screens seem to be incentivizing more stay at home behavior. People used to not own a TV, many quite intentionally, before our other screens were invented. But it is a very complicated topic.
Respectfully but strongly disagree. I'll argue you don't have to be a victim and can choose where you live if you plan ahead a little.
There's plenty of places where a car is not necessary and even if people think a car's necessary I'm often the only one on a bicycle in many places.. It's doable if you're willing to put in the effort.
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.