> If we want to lower cars in neighbourhoods, or want people to get rid of cars, our public transit system needs to be come a lot better first.

No, you can also start by imposing restrictions on cars.

> If public transit was a good option to get to work for people, more people would use it.

The people that live in the city center probably already use public transit because, for them, public transit is probably a faster commute. That means that the cars congesting the center are driven by people that live in the periphery or even suburbs outside of the city.

So these people will want those nasty cars out, because they actually live there, and they will vote them out.

You can never accommodate the people from the suburbs, unless they have a station right next to their door and their office and a train coming every 5 minutes, a car will always be faster. They live in someone else's municipality, so the mayors can just ignore them.

People in the periphery are more delicate, but usually they are also tired of congestions, and it's much easier to make at least minor improvements to public transit for them.

But again, you can restrict and make improvements in parallel, and the improvement will almost never be perceived as matching the restrictions anyway.