Create incentives for biking, but don't outlaw cars.
I do think cars should be banned in downtown-downtown though or at least have extremely harsh reinforcement: there is no reason for the rich businessman to be driving his Porsche at 120 km/h down a back road near where everyone cycles.
In my experience it's not the "rich businessman" but the "boy racer" with his Audi or VW Golf that are the most common problem.
In this way, cars can use the streets if they are indeed going to that area, but cannot move through the areas as a means to avoid roads meant to carry vehicles across town.
The city that exemplifies this is Groningen, Netherlands, and the superblock concept in Barcelona.
And if people start abusing that, at least their relatives or random elderly people get out and about for more sightseeing!
I know it causes issues. Not saying it can't be solved, but it is naive to say that simply banning cars is always a net plus. Why did people living in the city even buy cars to begin with?
My father used to be a doctor in a place that got converted into a pedestrians only area, he reported that a lot of people had issues. Business for other shops also declined.
(Please don't tell me about special bicycle seats.)
And we save costs due to less road maintenance (which is really expensive), no need to build new bridges/roads to solve traffic jams, lower health care costs, less time spent (i.e. money wasted) sitting in traffic jams, lower environmental costs, and you can put that in public infrastructure.
I think it's good to be mindful of these kind of issues: for all their downsides, cars can give great personal freedom and I think it's good to think how that can be maintained, but it's not like cars are the only way these problems can be solved.